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moocrr-reproducibility-study
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In this project, we gather reproduction attempts from the Challenger
study. In particular, we try to reperform some of the analysis provided
in *Risk Analysis of the Space Shuttle: Pre-Challenger Prediction of
Failure* by *Siddhartha R. Dalal, Edward B. Fowlkes, Bruce Hoadley*
published in *Journal of the American Statistical Association*, Vol. 84,
No. 408 (Dec., 1989), pp. 945-957 and available at
[here](https://studies2.hec.fr/jahia/webdav/site/hec/shared/sites/czellarv/acces_anonyme/OringJASA_1989.pdf)
(here is [the official JASA
webpage](http://www.jstor.org/stable/2290069)).
On the fourth page of this article, they indicate that the maximum
likelihood estimates of the logistic regression using only temperature
are: $`\hat{\alpha}=5.085`$ and $`\hat{\beta}=-0.1156`$ and their
asymptotic standard errors are $`s_{\hat{\alpha}}=3.052`$ and
$`s_{\hat{\beta}}=0.047`$. The Goodness of fit indicated for this model
was $`G^{2} = 18.086`$ with 21 degrees of freedom. Our goal is to
reproduce the computation behind these values and the Figure 4 of this
article, possibly in a nicer looking way.
[*Here is our successful replication of Dalal et al. results using
R*](challenger.pdf).
1. Try to **replicate the computation** from Dalal et al. In case it
helps, we provide you with twoimplementations of this case study
but we encourage you to **reimplement them by yourself** using both
your favourite language and an other language you do not know yet.
- A [Jupyter Python3 notebook](src/Python3/challenger.ipynb)
- An [Rmarkdown document](src/R/challenger.Rmd)
2. Then **update the following table with your own results by
indicating in each column:**
- Language: R, Python3, Julia, Perl, C...
- Language version:
- Main libraries: please indicate the versions of all the loaded
libraries
- Tool: Jupyter, Rstudio, Emacs
- Operating System: Linux, Mac OS X, Windows, Android, ... along with
its version
- $`\hat{\alpha}`$ and $`\hat{\beta}`$: Identical, Similar, Different,
Non functional (expected values are $`5.085`$ and $`-0.1156`$)
- $`s_{\hat{\alpha}}`$ and $`s_{\hat{\beta}}`$: Identical, Similar,
Different, Non functional (expected values are $`3.052`$ and $`0.047`$)
- $`G^2`$ and degree of freedom: Identical, Similar, Different, Non
functional (expected values are $`18.086`$ and $`21`$).
- Figure: Similar, Different, Non functional, Did not succeed
- Confidence region: Identical (to [the one obtained with R](challenger.pdf)), Similar, Quite Different, Did not succeed
| Language | Language version | Main libraries | Tool | Operating System | $`\hat{\alpha}= 5.085`$ $`\hat{\beta} = -0.1156`$ | $`s_{\hat{\alpha}} = 3.052`$ $`s_{\hat{\beta}} = 0.047`$ | $`G^{2} = 18.086`$ $`dof = 21`$ | Figure | Confidence Region | Link to the document | Author |
| -------- | ---------------- | ------------------------------------------------------------- | ------- | ---------------------------- | ------------- | ---------- | ---------- | --------- | ---------- | ----------------------------------------------------------------- | ----------- |
| R | 3.5.1 | ggplot2 3.0.0 | RStudio | Debian GNU/Linux buster/sid | Identical | Identical | Identical | Identical | Identical | [Rmd](src/R/challenger.Rmd), [pdf](src/R/challenger_debian_alegrand.pdf) | A. Legrand |
| Python | 3.6.4 | statsmodels 0.9.0 numpy 1.13.3 pandas 0.22.0 matplotlib 2.2.2 | Jupyter | Linux Ubuntu 4.4.0-116-generic | Identical | Identical | Identical | Identical | Similar | [ipynb](src/Python3/challenger.ipynb), [pdf](src/Python3/challenger_ubuntuMOOC_alegrand.pdf) | A. Legrand |
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Date,Count,Temperature,Pressure,Malfunction
4/12/81,6,66,50,0
11/12/81,6,70,50,1
3/22/82,6,69,50,0
11/11/82,6,68,50,0
4/04/83,6,67,50,0
6/18/82,6,72,50,0
8/30/83,6,73,100,0
11/28/83,6,70,100,0
2/03/84,6,57,200,1
4/06/84,6,63,200,1
8/30/84,6,70,200,1
10/05/84,6,78,200,0
11/08/84,6,67,200,0
1/24/85,6,53,200,2
4/12/85,6,67,200,0
4/29/85,6,75,200,0
6/17/85,6,70,200,0
7/2903/85,6,81,200,0
8/27/85,6,76,200,0
10/03/85,6,79,200,0
10/30/85,6,75,200,2
11/26/85,6,76,200,0
1/12/86,6,58,200,1
journal.md 0 → 100644
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# Journal
\ No newline at end of file
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# le titre de la premiere section
## le titre de la 2eme ss section
_italique_ *italique*
__gras__ **gras**
échasse fixeé 'chasse fixe' `chasse fixe`
--barré-- ~~barrer~~
un hyperline [Élaboration et conversion de documents avec Markdown et Pandoc](https://enacit1.epfl.ch/markdown-pandoc/)
<!-- un commentaire -->
![une photo de chat](https://upload.wikimedia.org/wikipedia/commons/thumb/6/64/Collage_of_Six_Cats-02.jpg/290px-Collage_of_Six_Cats-02.jpg)
liste à puce
- +1er élément
- 2 eme element
- 3 eme element
numérotation
1. premier
2. deuxieme
3. ...
Je peux imbriquer
1. ff
- a
- b
2. ggg
3. on continu
1. au début
2. ensuite
on va s'arrêter là
# Nouvelle Section de Niveau 1
- liste de Voitures
1. Porsche
2. Ferrari
- Liste d'éditeurs
1. Notepad++
2. TextEdit
## Nouvelle Section de Niveau 2
Bonjour, c'est ma nouvelle Section de niveau 2,
qui permet de créer un heading de niveau 2 en HTML.
### C'est un paragraphe formé de plusieurs lignes
<p> Ligne 1 du paragraphe <br> <hr>Ligne 2 du paragraphe <br> <hr> Ligne 3 du paragraphe
</p>
<br>
*Texte en Italique* <br>
__texte en gras__ <br>
___texte en italique et gras___
<br>
[Lien vers Google](https://www.google.fr/)
<br>
![Image ](http://animtoit.fr/exemple-chien/)
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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Risk Analysis of the Space Shuttle: Pre-Challenger Prediction of Failure"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In this document we reperform some of the analysis provided in \n",
"*Risk Analysis of the Space Shuttle: Pre-Challenger Prediction of Failure* by *Siddhartha R. Dalal, Edward B. Fowlkes, Bruce Hoadley* published in *Journal of the American Statistical Association*, Vol. 84, No. 408 (Dec., 1989), pp. 945-957 and available at http://www.jstor.org/stable/2290069. \n",
"\n",
"On the fourth page of this article, they indicate that the maximum likelihood estimates of the logistic regression using only temperature are: $\\hat{\\alpha}=5.085$ and $\\hat{\\beta}=-0.1156$ and their asymptotic standard errors are $s_{\\hat{\\alpha}}=3.052$ and $s_{\\hat{\\beta}}=0.047$. The Goodness of fit indicated for this model was $G^2=18.086$ with 21 degrees of freedom. Our goal is to reproduce the computation behind these values and the Figure 4 of this article, possibly in a nicer looking way."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Technical information on the computer on which the analysis is run"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We will be using the python3 language using the pandas, statsmodels, numpy, matplotlib and seaborn libraries."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"3.6.4 |Anaconda, Inc.| (default, Jan 16 2018, 18:10:19) \n",
"[GCC 7.2.0]\n",
"uname_result(system='Linux', node='3a716011d2b6', release='4.4.0-116-generic', version='#140-Ubuntu SMP Mon Feb 12 21:23:04 UTC 2018', machine='x86_64', processor='x86_64')\n",
"IPython 6.4.0\n",
"IPython.core.release 6.4.0\n",
"PIL 5.2.0\n",
"PIL.Image 5.2.0\n",
"PIL._version 5.2.0\n",
"_csv 1.0\n",
"_ctypes 1.1.0\n",
"_curses b'2.2'\n",
"decimal 1.70\n",
"argparse 1.1\n",
"backcall 0.1.0\n",
"cffi 1.11.5\n",
"csv 1.0\n",
"ctypes 1.1.0\n",
"cycler 0.10.0\n",
"dateutil 2.7.3\n",
"decimal 1.70\n",
"decorator 4.3.0\n",
"distutils 3.6.4\n",
"ipaddress 1.0\n",
"ipykernel 4.8.2\n",
"ipykernel._version 4.8.2\n",
"ipython_genutils 0.2.0\n",
"ipython_genutils._version 0.2.0\n",
"ipywidgets 7.2.1\n",
"ipywidgets._version 7.2.1\n",
"jedi 0.12.1\n",
"json 2.0.9\n",
"jupyter_client 5.2.3\n",
"jupyter_client._version 5.2.3\n",
"jupyter_core 4.4.0\n",
"jupyter_core.version 4.4.0\n",
"kiwisolver 1.0.1\n",
"logging 0.5.1.2\n",
"matplotlib 2.2.2\n",
"matplotlib.backends.backend_agg 2.2.2\n",
"numpy 1.13.3\n",
"numpy.core 1.13.3\n",
"numpy.core.multiarray 3.1\n",
"numpy.core.umath b'0.4.0'\n",
"numpy.lib 1.13.3\n",
"numpy.linalg._umath_linalg b'0.1.5'\n",
"numpy.matlib 1.13.3\n",
"optparse 1.5.3\n",
"pandas 0.22.0\n",
"_libjson 1.33\n",
"parso 0.3.0\n",
"patsy 0.5.0\n",
"patsy.version 0.5.0\n",
"pexpect 4.6.0\n",
"pickleshare 0.7.4\n",
"platform 1.0.8\n",
"prompt_toolkit 1.0.15\n",
"ptyprocess 0.6.0\n",
"pygments 2.2.0\n",
"pyparsing 2.2.0\n",
"pytz 2018.5\n",
"re 2.2.1\n",
"scipy 1.1.0\n",
"scipy._lib.decorator 4.0.5\n",
"scipy._lib.six 1.2.0\n",
"scipy.fftpack._fftpack b'$Revision: $'\n",
"scipy.fftpack.convolve b'$Revision: $'\n",
"scipy.integrate._dop b'$Revision: $'\n",
"scipy.integrate._ode $Id$\n",
"scipy.integrate._odepack 1.9 \n",
"scipy.integrate._quadpack 1.13 \n",
"scipy.integrate.lsoda b'$Revision: $'\n",
"scipy.integrate.vode b'$Revision: $'\n",
"scipy.interpolate._fitpack 1.7 \n",
"scipy.interpolate.dfitpack b'$Revision: $'\n",
"scipy.linalg 0.4.9\n",
"scipy.linalg._fblas b'$Revision: $'\n",
"scipy.linalg._flapack b'$Revision: $'\n",
"scipy.linalg._flinalg b'$Revision: $'\n",
"scipy.ndimage 2.0\n",
"scipy.optimize._cobyla b'$Revision: $'\n",
"scipy.optimize._lbfgsb b'$Revision: $'\n",
"scipy.optimize._minpack 1.10 \n",
"scipy.optimize._nnls b'$Revision: $'\n",
"scipy.optimize._slsqp b'$Revision: $'\n",
"scipy.optimize.minpack2 b'$Revision: $'\n",
"scipy.signal.spline 0.2\n",
"scipy.sparse.linalg.eigen.arpack._arpack b'$Revision: $'\n",
"scipy.sparse.linalg.isolve._iterative b'$Revision: $'\n",
"scipy.special.specfun b'$Revision: $'\n",
"scipy.stats.mvn b'$Revision: $'\n",
"scipy.stats.statlib b'$Revision: $'\n",
"seaborn 0.8.1\n",
"seaborn.external.husl 2.1.0\n",
"seaborn.external.six 1.10.0\n",
"six 1.11.0\n",
"statsmodels 0.9.0\n",
"statsmodels.__init__ 0.9.0\n",
"traitlets 4.3.2\n",
"traitlets._version 4.3.2\n",
"urllib.request 3.6\n",
"zlib 1.0\n",
"zmq 17.1.0\n",
"zmq.sugar 17.1.0\n",
"zmq.sugar.version 17.1.0\n"
]
}
],
"source": [
"def print_imported_modules():\n",
" import sys\n",
" for name, val in sorted(sys.modules.items()):\n",
" if(hasattr(val, '__version__')): \n",
" print(val.__name__, val.__version__)\n",
"# else:\n",
"# print(val.__name__, \"(unknown version)\")\n",
"def print_sys_info():\n",
" import sys\n",
" import platform\n",
" print(sys.version)\n",
" print(platform.uname())\n",
"\n",
"import numpy as np\n",
"import pandas as pd\n",
"import matplotlib.pyplot as plt\n",
"import statsmodels.api as sm\n",
"import seaborn as sns\n",
"\n",
"print_sys_info()\n",
"print_imported_modules()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Loading and inspecting data\n",
"Let's start by reading data."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<div>\n",
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"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>Date</th>\n",
" <th>Count</th>\n",
" <th>Temperature</th>\n",
" <th>Pressure</th>\n",
" <th>Malfunction</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>0</th>\n",
" <td>4/12/81</td>\n",
" <td>6</td>\n",
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" <td>50</td>\n",
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" <tr>\n",
" <th>2</th>\n",
" <td>3/22/82</td>\n",
" <td>6</td>\n",
" <td>69</td>\n",
" <td>50</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>3</th>\n",
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" <td>6</td>\n",
" <td>68</td>\n",
" <td>50</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
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" <td>6</td>\n",
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" <td>50</td>\n",
" <td>0</td>\n",
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" <th>5</th>\n",
" <td>6/18/82</td>\n",
" <td>6</td>\n",
" <td>72</td>\n",
" <td>50</td>\n",
" <td>0</td>\n",
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" <tr>\n",
" <th>6</th>\n",
" <td>8/30/83</td>\n",
" <td>6</td>\n",
" <td>73</td>\n",
" <td>100</td>\n",
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" <td>100</td>\n",
" <td>0</td>\n",
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" <td>2/03/84</td>\n",
" <td>6</td>\n",
" <td>57</td>\n",
" <td>200</td>\n",
" <td>1</td>\n",
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" <th>9</th>\n",
" <td>4/06/84</td>\n",
" <td>6</td>\n",
" <td>63</td>\n",
" <td>200</td>\n",
" <td>1</td>\n",
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" <td>200</td>\n",
" <td>1</td>\n",
" </tr>\n",
" <tr>\n",
" <th>11</th>\n",
" <td>10/05/84</td>\n",
" <td>6</td>\n",
" <td>78</td>\n",
" <td>200</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>12</th>\n",
" <td>11/08/84</td>\n",
" <td>6</td>\n",
" <td>67</td>\n",
" <td>200</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>13</th>\n",
" <td>1/24/85</td>\n",
" <td>6</td>\n",
" <td>53</td>\n",
" <td>200</td>\n",
" <td>2</td>\n",
" </tr>\n",
" <tr>\n",
" <th>14</th>\n",
" <td>4/12/85</td>\n",
" <td>6</td>\n",
" <td>67</td>\n",
" <td>200</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>15</th>\n",
" <td>4/29/85</td>\n",
" <td>6</td>\n",
" <td>75</td>\n",
" <td>200</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>16</th>\n",
" <td>6/17/85</td>\n",
" <td>6</td>\n",
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" <td>200</td>\n",
" <td>0</td>\n",
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" <tr>\n",
" <th>17</th>\n",
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" <td>6</td>\n",
" <td>81</td>\n",
" <td>200</td>\n",
" <td>0</td>\n",
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" <tr>\n",
" <th>18</th>\n",
" <td>8/27/85</td>\n",
" <td>6</td>\n",
" <td>76</td>\n",
" <td>200</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>19</th>\n",
" <td>10/03/85</td>\n",
" <td>6</td>\n",
" <td>79</td>\n",
" <td>200</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>20</th>\n",
" <td>10/30/85</td>\n",
" <td>6</td>\n",
" <td>75</td>\n",
" <td>200</td>\n",
" <td>2</td>\n",
" </tr>\n",
" <tr>\n",
" <th>21</th>\n",
" <td>11/26/85</td>\n",
" <td>6</td>\n",
" <td>76</td>\n",
" <td>200</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>22</th>\n",
" <td>1/12/86</td>\n",
" <td>6</td>\n",
" <td>58</td>\n",
" <td>200</td>\n",
" <td>1</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" Date Count Temperature Pressure Malfunction\n",
"0 4/12/81 6 66 50 0\n",
"1 11/12/81 6 70 50 1\n",
"2 3/22/82 6 69 50 0\n",
"3 11/11/82 6 68 50 0\n",
"4 4/04/83 6 67 50 0\n",
"5 6/18/82 6 72 50 0\n",
"6 8/30/83 6 73 100 0\n",
"7 11/28/83 6 70 100 0\n",
"8 2/03/84 6 57 200 1\n",
"9 4/06/84 6 63 200 1\n",
"10 8/30/84 6 70 200 1\n",
"11 10/05/84 6 78 200 0\n",
"12 11/08/84 6 67 200 0\n",
"13 1/24/85 6 53 200 2\n",
"14 4/12/85 6 67 200 0\n",
"15 4/29/85 6 75 200 0\n",
"16 6/17/85 6 70 200 0\n",
"17 7/2903/85 6 81 200 0\n",
"18 8/27/85 6 76 200 0\n",
"19 10/03/85 6 79 200 0\n",
"20 10/30/85 6 75 200 2\n",
"21 11/26/85 6 76 200 0\n",
"22 1/12/86 6 58 200 1"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"data = pd.read_csv(\"https://app-learninglab.inria.fr/gitlab/moocrr-session1/moocrr-reproducibility-study/raw/master/data/shuttle.csv\")\n",
"data"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We know from our previous experience on this data set that filtering data is a really bad idea. We will therefore process it as such."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%matplotlib inline\n",
"pd.set_option('mode.chained_assignment',None) # this removes a useless warning from pandas\n",
"import matplotlib.pyplot as plt\n",
"\n",
"data[\"Frequency\"]=data.Malfunction/data.Count\n",
"data.plot(x=\"Temperature\",y=\"Frequency\",kind=\"scatter\",ylim=[0,1])\n",
"plt.grid(True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Logistic regression\n",
"\n",
"Let's assume O-rings independently fail with the same probability which solely depends on temperature. A logistic regression should allow us to estimate the influence of temperature."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<table class=\"simpletable\">\n",
"<caption>Generalized Linear Model Regression Results</caption>\n",
"<tr>\n",
" <th>Dep. Variable:</th> <td>Frequency</td> <th> No. Observations: </th> <td> 23</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Model:</th> <td>GLM</td> <th> Df Residuals: </th> <td> 21</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Model Family:</th> <td>Binomial</td> <th> Df Model: </th> <td> 1</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Link Function:</th> <td>logit</td> <th> Scale: </th> <td> 1.0000</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Method:</th> <td>IRLS</td> <th> Log-Likelihood: </th> <td> -3.9210</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Date:</th> <td>Wed, 24 Oct 2018</td> <th> Deviance: </th> <td> 3.0144</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Time:</th> <td>11:05:55</td> <th> Pearson chi2: </th> <td> 5.00</td> \n",
"</tr>\n",
"<tr>\n",
" <th>No. Iterations:</th> <td>6</td> <th> Covariance Type: </th> <td>nonrobust</td>\n",
"</tr>\n",
"</table>\n",
"<table class=\"simpletable\">\n",
"<tr>\n",
" <td></td> <th>coef</th> <th>std err</th> <th>z</th> <th>P>|z|</th> <th>[0.025</th> <th>0.975]</th> \n",
"</tr>\n",
"<tr>\n",
" <th>Intercept</th> <td> 5.0850</td> <td> 7.477</td> <td> 0.680</td> <td> 0.496</td> <td> -9.570</td> <td> 19.740</td>\n",
"</tr>\n",
"<tr>\n",
" <th>Temperature</th> <td> -0.1156</td> <td> 0.115</td> <td> -1.004</td> <td> 0.316</td> <td> -0.341</td> <td> 0.110</td>\n",
"</tr>\n",
"</table>"
],
"text/plain": [
"<class 'statsmodels.iolib.summary.Summary'>\n",
"\"\"\"\n",
" Generalized Linear Model Regression Results \n",
"==============================================================================\n",
"Dep. Variable: Frequency No. Observations: 23\n",
"Model: GLM Df Residuals: 21\n",
"Model Family: Binomial Df Model: 1\n",
"Link Function: logit Scale: 1.0000\n",
"Method: IRLS Log-Likelihood: -3.9210\n",
"Date: Wed, 24 Oct 2018 Deviance: 3.0144\n",
"Time: 11:05:55 Pearson chi2: 5.00\n",
"No. Iterations: 6 Covariance Type: nonrobust\n",
"===============================================================================\n",
" coef std err z P>|z| [0.025 0.975]\n",
"-------------------------------------------------------------------------------\n",
"Intercept 5.0850 7.477 0.680 0.496 -9.570 19.740\n",
"Temperature -0.1156 0.115 -1.004 0.316 -0.341 0.110\n",
"===============================================================================\n",
"\"\"\""
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import statsmodels.api as sm\n",
"\n",
"data[\"Success\"]=data.Count-data.Malfunction\n",
"data[\"Intercept\"]=1\n",
"\n",
"logmodel=sm.GLM(data['Frequency'], data[['Intercept','Temperature']], \n",
" family=sm.families.Binomial(sm.families.links.logit)).fit()\n",
"\n",
"logmodel.summary()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The maximum likelyhood estimator of the intercept and of Temperature are thus $\\hat{\\alpha}=5.0849$ and $\\hat{\\beta}=-0.1156$. This **corresponds** to the values from the article of Dalal *et al.* The standard errors are $s_{\\hat{\\alpha}} = 7.477$ and $s_{\\hat{\\beta}} = 0.115$, which is **different** from the $3.052$ and $0.04702$ reported by Dallal *et al.* The deviance is $3.01444$ with 21 degrees of freedom. I cannot find any value similar to the Goodness of fit ($G^2=18.086$) reported by Dalal *et al.* There seems to be something wrong. Oh I know, I haven't indicated that my observations are actually the result of 6 observations for each rocket launch. Let's indicate these weights (since the weights are always the same throughout all experiments, it does not change the estimates of the fit but it does influence the variance estimates)."
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<table class=\"simpletable\">\n",
"<caption>Generalized Linear Model Regression Results</caption>\n",
"<tr>\n",
" <th>Dep. Variable:</th> <td>Frequency</td> <th> No. Observations: </th> <td> 23</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Model:</th> <td>GLM</td> <th> Df Residuals: </th> <td> 21</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Model Family:</th> <td>Binomial</td> <th> Df Model: </th> <td> 1</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Link Function:</th> <td>logit</td> <th> Scale: </th> <td> 1.0000</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Method:</th> <td>IRLS</td> <th> Log-Likelihood: </th> <td> -23.526</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Date:</th> <td>Wed, 24 Oct 2018</td> <th> Deviance: </th> <td> 18.086</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Time:</th> <td>11:05:55</td> <th> Pearson chi2: </th> <td> 30.0</td> \n",
"</tr>\n",
"<tr>\n",
" <th>No. Iterations:</th> <td>6</td> <th> Covariance Type: </th> <td>nonrobust</td>\n",
"</tr>\n",
"</table>\n",
"<table class=\"simpletable\">\n",
"<tr>\n",
" <td></td> <th>coef</th> <th>std err</th> <th>z</th> <th>P>|z|</th> <th>[0.025</th> <th>0.975]</th> \n",
"</tr>\n",
"<tr>\n",
" <th>Intercept</th> <td> 5.0850</td> <td> 3.052</td> <td> 1.666</td> <td> 0.096</td> <td> -0.898</td> <td> 11.068</td>\n",
"</tr>\n",
"<tr>\n",
" <th>Temperature</th> <td> -0.1156</td> <td> 0.047</td> <td> -2.458</td> <td> 0.014</td> <td> -0.208</td> <td> -0.023</td>\n",
"</tr>\n",
"</table>"
],
"text/plain": [
"<class 'statsmodels.iolib.summary.Summary'>\n",
"\"\"\"\n",
" Generalized Linear Model Regression Results \n",
"==============================================================================\n",
"Dep. Variable: Frequency No. Observations: 23\n",
"Model: GLM Df Residuals: 21\n",
"Model Family: Binomial Df Model: 1\n",
"Link Function: logit Scale: 1.0000\n",
"Method: IRLS Log-Likelihood: -23.526\n",
"Date: Wed, 24 Oct 2018 Deviance: 18.086\n",
"Time: 11:05:55 Pearson chi2: 30.0\n",
"No. Iterations: 6 Covariance Type: nonrobust\n",
"===============================================================================\n",
" coef std err z P>|z| [0.025 0.975]\n",
"-------------------------------------------------------------------------------\n",
"Intercept 5.0850 3.052 1.666 0.096 -0.898 11.068\n",
"Temperature -0.1156 0.047 -2.458 0.014 -0.208 -0.023\n",
"===============================================================================\n",
"\"\"\""
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"logmodel=sm.GLM(data['Frequency'], data[['Intercept','Temperature']], \n",
" family=sm.families.Binomial(sm.families.links.logit),\n",
" var_weights=data['Count']).fit()\n",
"\n",
"logmodel.summary()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Good, now I have recovered the asymptotic standard errors $s_{\\hat{\\alpha}}=3.052$ and $s_{\\hat{\\beta}}=0.047$.\n",
"The Goodness of fit (Deviance) indicated for this model is $G^2=18.086$ with 21 degrees of freedom (Df Residuals).\n",
"\n",
"**I have therefore managed to fully replicate the results of the Dalal *et al.* article**."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Predicting failure probability\n",
"The temperature when launching the shuttle was 31°F. Let's try to estimate the failure probability for such temperature using our model.:"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%matplotlib inline\n",
"data_pred = pd.DataFrame({'Temperature': np.linspace(start=30, stop=90, num=121), 'Intercept': 1})\n",
"data_pred['Frequency'] = logmodel.predict(data_pred)\n",
"data_pred.plot(x=\"Temperature\",y=\"Frequency\",kind=\"line\",ylim=[0,1])\n",
"plt.scatter(x=data[\"Temperature\"],y=data[\"Frequency\"])\n",
"plt.grid(True)"
]
},
{
"cell_type": "markdown",
"metadata": {
"hideCode": false,
"hidePrompt": false,
"scrolled": true
},
"source": [
"This figure is very similar to the Figure 4 of Dalal *et al.* **I have managed to replicate the Figure 4 of the Dalal *et al.* article.**"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Computing and plotting uncertainty"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Following the documentation of [Seaborn](https://seaborn.pydata.org/generated/seaborn.regplot.html), I use regplot."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"sns.set(color_codes=True)\n",
"plt.xlim(30,90)\n",
"plt.ylim(0,1)\n",
"sns.regplot(x='Temperature', y='Frequency', data=data, logistic=True)\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**I think I have managed to correctly compute and plot the uncertainty of my prediction.** Although the shaded area seems very similar to [the one obtained by with R](https://app-learninglab.inria.fr/gitlab/moocrr-session1/moocrr-reproducibility-study/raw/5c9dbef11b4d7638b7ddf2ea71026e7bf00fcfb0/challenger.pdf), I can spot a few differences (e.g., the blue point for temperature 63 is outside)... Could this be a numerical error ? Or a difference in the statistical method ? It is not clear which one is \"right\"."
]
}
],
"metadata": {
"celltoolbar": "Hide code",
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.6.6"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
src/Python3/challenger_Python_ipynb.ipynb 0 → 100644
View file @ 9d85e5f4
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Risk Analysis of the Space Shuttle: Pre-Challenger Prediction of Failure"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In this document we reperform some of the analysis provided in \n",
"*Risk Analysis of the Space Shuttle: Pre-Challenger Prediction of Failure* by *Siddhartha R. Dalal, Edward B. Fowlkes, Bruce Hoadley* published in *Journal of the American Statistical Association*, Vol. 84, No. 408 (Dec., 1989), pp. 945-957 and available at http://www.jstor.org/stable/2290069. \n",
"\n",
"On the fourth page of this article, they indicate that the maximum likelihood estimates of the logistic regression using only temperature are: $\\hat{\\alpha}$ = **5.085** and $\\hat{\\beta}$ = **-0.1156** and their asymptotic standard errors are $s_{\\hat{\\alpha}}$ = **3.052** and $s_{\\hat{\\beta}}$ = **0.047**. The Goodness of fit indicated for this model was $G^2$ = **18.086** with **21** degrees of freedom. Our goal is to reproduce the computation behind these values and the Figure 4 of this article, possibly in a nicer looking way."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Technical information on the computer on which the analysis is run"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We will be using the Python 3 language using the pandas, statsmodels, and numpy library."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"3.7.0 (v3.7.0:1bf9cc5093, Jun 27 2018, 04:59:51) [MSC v.1914 64 bit (AMD64)]\n",
"uname_result(system='Windows', node='MGDONDON', release='7', version='6.1.7601', machine='AMD64', processor='Intel64 Family 6 Model 94 Stepping 3, GenuineIntel')\n",
"IPython 6.5.0\n",
"IPython.core.release 6.5.0\n",
"_csv 1.0\n",
"_ctypes 1.1.0\n",
"decimal 1.70\n",
"argparse 1.1\n",
"backcall 0.1.0\n",
"colorama 0.3.9\n",
"csv 1.0\n",
"ctypes 1.1.0\n",
"cycler 0.10.0\n",
"dateutil 2.7.3\n",
"decimal 1.70\n",
"decorator 4.3.0\n",
"distutils 3.7.0\n",
"ipykernel 4.8.2\n",
"ipykernel._version 4.8.2\n",
"ipython_genutils 0.2.0\n",
"ipython_genutils._version 0.2.0\n",
"ipywidgets 7.4.0\n",
"ipywidgets._version 7.4.0\n",
"jedi 0.12.1\n",
"json 2.0.9\n",
"jupyter_client 5.2.3\n",
"jupyter_client._version 5.2.3\n",
"jupyter_core 4.4.0\n",
"jupyter_core.version 4.4.0\n",
"kiwisolver 1.0.1\n",
"logging 0.5.1.2\n",
"matplotlib 2.2.3\n",
"matplotlib.backends.backend_agg 2.2.3\n",
"numpy 1.15.4\n",
"numpy.core 1.15.4\n",
"numpy.core.multiarray 3.1\n",
"numpy.lib 1.15.4\n",
"numpy.linalg._umath_linalg b'0.1.5'\n",
"numpy.matlib 1.15.4\n",
"pandas 0.23.4\n",
"_libjson 1.33\n",
"parso 0.3.1\n",
"patsy 0.5.0\n",
"patsy.version 0.5.0\n",
"pickleshare 0.7.4\n",
"platform 1.0.8\n",
"prompt_toolkit 1.0.15\n",
"pygments 2.2.0\n",
"pyparsing 2.2.0\n",
"pytz 2018.5\n",
"re 2.2.1\n",
"scipy 1.1.0\n",
"scipy._lib.decorator 4.0.5\n",
"scipy._lib.six 1.2.0\n",
"scipy.fftpack._fftpack b'$Revision: $'\n",
"scipy.fftpack.convolve b'$Revision: $'\n",
"scipy.integrate._dop b'$Revision: $'\n",
"scipy.integrate._ode $Id$\n",
"scipy.integrate._odepack 1.9 \n",
"scipy.integrate._quadpack 1.13 \n",
"scipy.integrate.lsoda b'$Revision: $'\n",
"scipy.integrate.vode b'$Revision: $'\n",
"scipy.interpolate._fitpack 1.7 \n",
"scipy.interpolate.dfitpack b'$Revision: $'\n",
"scipy.linalg 0.4.9\n",
"scipy.linalg._fblas b'$Revision: $'\n",
"scipy.linalg._flapack b'$Revision: $'\n",
"scipy.linalg._flinalg b'$Revision: $'\n",
"scipy.ndimage 2.0\n",
"scipy.optimize._cobyla b'$Revision: $'\n",
"scipy.optimize._lbfgsb b'$Revision: $'\n",
"scipy.optimize._minpack 1.10 \n",
"scipy.optimize._nnls b'$Revision: $'\n",
"scipy.optimize._slsqp b'$Revision: $'\n",
"scipy.optimize.minpack2 b'$Revision: $'\n",
"scipy.signal.spline 0.2\n",
"scipy.sparse.linalg.eigen.arpack._arpack b'$Revision: $'\n",
"scipy.sparse.linalg.isolve._iterative b'$Revision: $'\n",
"scipy.special.specfun b'$Revision: $'\n",
"scipy.stats.mvn b'$Revision: $'\n",
"scipy.stats.statlib b'$Revision: $'\n",
"seaborn 0.9.0\n",
"seaborn.external.husl 2.1.0\n",
"seaborn.external.six 1.10.0\n",
"six 1.11.0\n",
"statsmodels 0.9.0\n",
"statsmodels.__init__ 0.9.0\n",
"traitlets 4.3.2\n",
"traitlets._version 4.3.2\n",
"urllib.request 3.7\n",
"zlib 1.0\n",
"zmq 17.1.2\n",
"zmq.sugar 17.1.2\n",
"zmq.sugar.version 17.1.2\n"
]
}
],
"source": [
"def print_imported_modules():\n",
" import sys\n",
" for name, val in sorted(sys.modules.items()):\n",
" if(hasattr(val, '__version__')): \n",
" print(val.__name__, val.__version__)\n",
"# else:\n",
"# print(val.__name__, \"(unknown version)\")\n",
"def print_sys_info():\n",
" import sys\n",
" import platform\n",
" print(sys.version)\n",
" print(platform.uname())\n",
"\n",
"import numpy as np\n",
"import pandas as pd\n",
"import matplotlib.pyplot as plt\n",
"import statsmodels.api as sm\n",
"import seaborn as sns\n",
"\n",
"print_sys_info()\n",
"print_imported_modules()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Loading and inspecting data\n",
"Let's start by reading data."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<div>\n",
"<style scoped>\n",
" .dataframe tbody tr th:only-of-type {\n",
" vertical-align: middle;\n",
" }\n",
"\n",
" .dataframe tbody tr th {\n",
" vertical-align: top;\n",
" }\n",
"\n",
" .dataframe thead th {\n",
" text-align: right;\n",
" }\n",
"</style>\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>Date</th>\n",
" <th>Count</th>\n",
" <th>Temperature</th>\n",
" <th>Pressure</th>\n",
" <th>Malfunction</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>0</th>\n",
" <td>4/12/81</td>\n",
" <td>6</td>\n",
" <td>66</td>\n",
" <td>50</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>1</th>\n",
" <td>11/12/81</td>\n",
" <td>6</td>\n",
" <td>70</td>\n",
" <td>50</td>\n",
" <td>1</td>\n",
" </tr>\n",
" <tr>\n",
" <th>2</th>\n",
" <td>3/22/82</td>\n",
" <td>6</td>\n",
" <td>69</td>\n",
" <td>50</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>3</th>\n",
" <td>11/11/82</td>\n",
" <td>6</td>\n",
" <td>68</td>\n",
" <td>50</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>4</th>\n",
" <td>4/04/83</td>\n",
" <td>6</td>\n",
" <td>67</td>\n",
" <td>50</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>5</th>\n",
" <td>6/18/82</td>\n",
" <td>6</td>\n",
" <td>72</td>\n",
" <td>50</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>6</th>\n",
" <td>8/30/83</td>\n",
" <td>6</td>\n",
" <td>73</td>\n",
" <td>100</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>7</th>\n",
" <td>11/28/83</td>\n",
" <td>6</td>\n",
" <td>70</td>\n",
" <td>100</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>8</th>\n",
" <td>2/03/84</td>\n",
" <td>6</td>\n",
" <td>57</td>\n",
" <td>200</td>\n",
" <td>1</td>\n",
" </tr>\n",
" <tr>\n",
" <th>9</th>\n",
" <td>4/06/84</td>\n",
" <td>6</td>\n",
" <td>63</td>\n",
" <td>200</td>\n",
" <td>1</td>\n",
" </tr>\n",
" <tr>\n",
" <th>10</th>\n",
" <td>8/30/84</td>\n",
" <td>6</td>\n",
" <td>70</td>\n",
" <td>200</td>\n",
" <td>1</td>\n",
" </tr>\n",
" <tr>\n",
" <th>11</th>\n",
" <td>10/05/84</td>\n",
" <td>6</td>\n",
" <td>78</td>\n",
" <td>200</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>12</th>\n",
" <td>11/08/84</td>\n",
" <td>6</td>\n",
" <td>67</td>\n",
" <td>200</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>13</th>\n",
" <td>1/24/85</td>\n",
" <td>6</td>\n",
" <td>53</td>\n",
" <td>200</td>\n",
" <td>2</td>\n",
" </tr>\n",
" <tr>\n",
" <th>14</th>\n",
" <td>4/12/85</td>\n",
" <td>6</td>\n",
" <td>67</td>\n",
" <td>200</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>15</th>\n",
" <td>4/29/85</td>\n",
" <td>6</td>\n",
" <td>75</td>\n",
" <td>200</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>16</th>\n",
" <td>6/17/85</td>\n",
" <td>6</td>\n",
" <td>70</td>\n",
" <td>200</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>17</th>\n",
" <td>7/2903/85</td>\n",
" <td>6</td>\n",
" <td>81</td>\n",
" <td>200</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>18</th>\n",
" <td>8/27/85</td>\n",
" <td>6</td>\n",
" <td>76</td>\n",
" <td>200</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>19</th>\n",
" <td>10/03/85</td>\n",
" <td>6</td>\n",
" <td>79</td>\n",
" <td>200</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>20</th>\n",
" <td>10/30/85</td>\n",
" <td>6</td>\n",
" <td>75</td>\n",
" <td>200</td>\n",
" <td>2</td>\n",
" </tr>\n",
" <tr>\n",
" <th>21</th>\n",
" <td>11/26/85</td>\n",
" <td>6</td>\n",
" <td>76</td>\n",
" <td>200</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>22</th>\n",
" <td>1/12/86</td>\n",
" <td>6</td>\n",
" <td>58</td>\n",
" <td>200</td>\n",
" <td>1</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" Date Count Temperature Pressure Malfunction\n",
"0 4/12/81 6 66 50 0\n",
"1 11/12/81 6 70 50 1\n",
"2 3/22/82 6 69 50 0\n",
"3 11/11/82 6 68 50 0\n",
"4 4/04/83 6 67 50 0\n",
"5 6/18/82 6 72 50 0\n",
"6 8/30/83 6 73 100 0\n",
"7 11/28/83 6 70 100 0\n",
"8 2/03/84 6 57 200 1\n",
"9 4/06/84 6 63 200 1\n",
"10 8/30/84 6 70 200 1\n",
"11 10/05/84 6 78 200 0\n",
"12 11/08/84 6 67 200 0\n",
"13 1/24/85 6 53 200 2\n",
"14 4/12/85 6 67 200 0\n",
"15 4/29/85 6 75 200 0\n",
"16 6/17/85 6 70 200 0\n",
"17 7/2903/85 6 81 200 0\n",
"18 8/27/85 6 76 200 0\n",
"19 10/03/85 6 79 200 0\n",
"20 10/30/85 6 75 200 2\n",
"21 11/26/85 6 76 200 0\n",
"22 1/12/86 6 58 200 1"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"data = pd.read_csv(\"https://app-learninglab.inria.fr/gitlab/moocrr-session1/moocrr-reproducibility-study/raw/master/data/shuttle.csv\")\n",
"data"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We know from our previous experience on this data set that filtering data is a really bad idea. We will therefore process it as such."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%matplotlib inline\n",
"pd.set_option('mode.chained_assignment',None) # this removes a useless warning from pandas\n",
"import matplotlib.pyplot as plt\n",
"\n",
"data[\"Frequency\"]=data.Malfunction/data.Count\n",
"data.plot(x=\"Temperature\",y=\"Frequency\",kind=\"scatter\",ylim=[0,1])\n",
"plt.grid(True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Logistic regression\n",
"\n",
"Let's assume O-rings independently fail with the same probability which solely depends on temperature. A logistic regression should allow us to estimate the influence of temperature."
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<table class=\"simpletable\">\n",
"<caption>Generalized Linear Model Regression Results</caption>\n",
"<tr>\n",
" <th>Dep. Variable:</th> <td>Frequency</td> <th> No. Observations: </th> <td> 23</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Model:</th> <td>GLM</td> <th> Df Residuals: </th> <td> 21</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Model Family:</th> <td>Binomial</td> <th> Df Model: </th> <td> 1</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Link Function:</th> <td>logit</td> <th> Scale: </th> <td> 1.0000</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Method:</th> <td>IRLS</td> <th> Log-Likelihood: </th> <td> -3.9210</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Date:</th> <td>Mon, 12 Nov 2018</td> <th> Deviance: </th> <td> 3.0144</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Time:</th> <td>21:08:43</td> <th> Pearson chi2: </th> <td> 5.00</td> \n",
"</tr>\n",
"<tr>\n",
" <th>No. Iterations:</th> <td>6</td> <th> Covariance Type: </th> <td>nonrobust</td>\n",
"</tr>\n",
"</table>\n",
"<table class=\"simpletable\">\n",
"<tr>\n",
" <td></td> <th>coef</th> <th>std err</th> <th>z</th> <th>P>|z|</th> <th>[0.025</th> <th>0.975]</th> \n",
"</tr>\n",
"<tr>\n",
" <th>Intercept</th> <td> 5.0850</td> <td> 7.477</td> <td> 0.680</td> <td> 0.496</td> <td> -9.570</td> <td> 19.740</td>\n",
"</tr>\n",
"<tr>\n",
" <th>Temperature</th> <td> -0.1156</td> <td> 0.115</td> <td> -1.004</td> <td> 0.316</td> <td> -0.341</td> <td> 0.110</td>\n",
"</tr>\n",
"</table>"
],
"text/plain": [
"<class 'statsmodels.iolib.summary.Summary'>\n",
"\"\"\"\n",
" Generalized Linear Model Regression Results \n",
"==============================================================================\n",
"Dep. Variable: Frequency No. Observations: 23\n",
"Model: GLM Df Residuals: 21\n",
"Model Family: Binomial Df Model: 1\n",
"Link Function: logit Scale: 1.0000\n",
"Method: IRLS Log-Likelihood: -3.9210\n",
"Date: Mon, 12 Nov 2018 Deviance: 3.0144\n",
"Time: 21:08:43 Pearson chi2: 5.00\n",
"No. Iterations: 6 Covariance Type: nonrobust\n",
"===============================================================================\n",
" coef std err z P>|z| [0.025 0.975]\n",
"-------------------------------------------------------------------------------\n",
"Intercept 5.0850 7.477 0.680 0.496 -9.570 19.740\n",
"Temperature -0.1156 0.115 -1.004 0.316 -0.341 0.110\n",
"===============================================================================\n",
"\"\"\""
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import statsmodels.api as sm\n",
"\n",
"data[\"Success\"]=data.Count-data.Malfunction\n",
"data[\"Intercept\"]=1\n",
"\n",
"logmodel=sm.GLM(data['Frequency'], data[['Intercept','Temperature']], \n",
" family=sm.families.Binomial(sm.families.links.logit)).fit()\n",
"\n",
"logmodel.summary()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The maximum likelyhood estimator of the intercept and of Temperature are thus $\\hat{\\alpha}$ = **5.0850** and $\\hat{\\beta}$ = **-0.1156**. This **corresponds** to the values from the article of Dalal *et al.* The standard errors are $s_{\\hat{\\alpha}}$ = ***7.477*** and $s_{\\hat{\\beta}}$ = ***0.115***, which is **different** from the **3.052** and **0.04702** reported by Dallal *et al.* The deviance is ***3.01444*** with **21** degrees of freedom. I cannot find any value similar to the Goodness of fit ($G^2$ = **18.086**) reported by Dalal *et al.* There seems to be something wrong. Oh I know, I haven't indicated that my observations are actually the result of 6 observations for each rocket launch. Let's indicate these weights (since the weights are always the same throughout all experiments, it does not change the estimates of the fit but it does influence the variance estimates)."
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<table class=\"simpletable\">\n",
"<caption>Generalized Linear Model Regression Results</caption>\n",
"<tr>\n",
" <th>Dep. Variable:</th> <td>Frequency</td> <th> No. Observations: </th> <td> 23</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Model:</th> <td>GLM</td> <th> Df Residuals: </th> <td> 21</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Model Family:</th> <td>Binomial</td> <th> Df Model: </th> <td> 1</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Link Function:</th> <td>logit</td> <th> Scale: </th> <td> 1.0000</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Method:</th> <td>IRLS</td> <th> Log-Likelihood: </th> <td> -23.526</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Date:</th> <td>Mon, 12 Nov 2018</td> <th> Deviance: </th> <td> 18.086</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Time:</th> <td>21:08:47</td> <th> Pearson chi2: </th> <td> 30.0</td> \n",
"</tr>\n",
"<tr>\n",
" <th>No. Iterations:</th> <td>6</td> <th> Covariance Type: </th> <td>nonrobust</td>\n",
"</tr>\n",
"</table>\n",
"<table class=\"simpletable\">\n",
"<tr>\n",
" <td></td> <th>coef</th> <th>std err</th> <th>z</th> <th>P>|z|</th> <th>[0.025</th> <th>0.975]</th> \n",
"</tr>\n",
"<tr>\n",
" <th>Intercept</th> <td> 5.0850</td> <td> 3.052</td> <td> 1.666</td> <td> 0.096</td> <td> -0.898</td> <td> 11.068</td>\n",
"</tr>\n",
"<tr>\n",
" <th>Temperature</th> <td> -0.1156</td> <td> 0.047</td> <td> -2.458</td> <td> 0.014</td> <td> -0.208</td> <td> -0.023</td>\n",
"</tr>\n",
"</table>"
],
"text/plain": [
"<class 'statsmodels.iolib.summary.Summary'>\n",
"\"\"\"\n",
" Generalized Linear Model Regression Results \n",
"==============================================================================\n",
"Dep. Variable: Frequency No. Observations: 23\n",
"Model: GLM Df Residuals: 21\n",
"Model Family: Binomial Df Model: 1\n",
"Link Function: logit Scale: 1.0000\n",
"Method: IRLS Log-Likelihood: -23.526\n",
"Date: Mon, 12 Nov 2018 Deviance: 18.086\n",
"Time: 21:08:47 Pearson chi2: 30.0\n",
"No. Iterations: 6 Covariance Type: nonrobust\n",
"===============================================================================\n",
" coef std err z P>|z| [0.025 0.975]\n",
"-------------------------------------------------------------------------------\n",
"Intercept 5.0850 3.052 1.666 0.096 -0.898 11.068\n",
"Temperature -0.1156 0.047 -2.458 0.014 -0.208 -0.023\n",
"===============================================================================\n",
"\"\"\""
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"logmodel=sm.GLM(data['Frequency'], data[['Intercept','Temperature']], \n",
" family=sm.families.Binomial(sm.families.links.logit),\n",
" var_weights=data['Count']).fit()\n",
"\n",
"logmodel.summary()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Good, now I have recovered the asymptotic standard errors $s_{\\hat{\\alpha}}$ = **3.052** and $s_{\\hat{\\beta}}$ = **0.047**.\n",
"The Goodness of fit (Deviance) indicated for this model is $G^2$ = **18.086** with **21** degrees of freedom (Df Residuals).\n",
"\n",
"**I have therefore managed to fully replicate the results of the Dalal *et al.* article**."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Predicting failure probability\n",
"The temperature when launching the shuttle was 31°F. Let's try to estimate the failure probability for such temperature using our model:"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" Temperature Intercept Frequency\n",
"0 30.0 1 1.0\n",
"1 30.5 1 1.0\n",
"2 31.0 1 1.0\n",
"3 31.5 1 1.0\n",
"4 32.0 1 1.0\n"
]
}
],
"source": [
"data_pred = pd.DataFrame({'Temperature': np.linspace(start=30, stop=90, num=121),\n",
" 'Intercept': 1})\n",
"data_pred['Frequency'] = logmodel.predict(data_pred)\n",
"print(data_pred.head())"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%matplotlib inline\n",
"data_pred.plot(x=\"Temperature\",y=\"Frequency\",kind=\"line\",ylim=[0,1])\n",
"plt.scatter(x=data[\"Temperature\"],y=data[\"Frequency\"])\n",
"plt.grid(True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"La fonction `logmodel.predict(data_pred)` ne fonctionne pas avec les dernières versions de pandas (`Frequency = 1` pour toutes les températures).\n",
"\n",
"On peut alors utiliser le code suivant pour calculer les prédictions et tracer la courbe :"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" Temperature Intercept Frequency Prob\n",
"0 30.0 1 1.0 0.834373\n",
"1 30.5 1 1.0 0.826230\n",
"2 31.0 1 1.0 0.817774\n",
"3 31.5 1 1.0 0.809002\n",
"4 32.0 1 1.0 0.799911\n"
]
}
],
"source": [
"# Inspiring from http://blog.yhat.com/posts/logistic-regression-and-python.html\n",
"def logit_inv(x):\n",
" return(np.exp(x)/(np.exp(x)+1))\n",
"\n",
"data_pred['Prob']=logit_inv(data_pred['Temperature'] * logmodel.params['Temperature'] + \n",
" logmodel.params['Intercept'])\n",
"print(data_pred.head())"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%matplotlib inline\n",
"data_pred.plot(x=\"Temperature\",y=\"Prob\",kind=\"line\",ylim=[0,1])\n",
"plt.scatter(x=data[\"Temperature\"],y=data[\"Frequency\"])\n",
"plt.grid(True)"
]
},
{
"cell_type": "markdown",
"metadata": {
"hideCode": false,
"hidePrompt": false,
"scrolled": true
},
"source": [
"This figure is very similar to the Figure 4 of Dalal *et al.* **I have managed to replicate the Figure 4 of the Dalal *et al.* article.**"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Computing and plotting uncertainty"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Following the documentation of [Seaborn](https://seaborn.pydata.org/generated/seaborn.regplot.html), I use regplot."
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"c:\\program files\\python\\python37\\lib\\site-packages\\scipy\\stats\\stats.py:1713: FutureWarning: Using a non-tuple sequence for multidimensional indexing is deprecated; use `arr[tuple(seq)]` instead of `arr[seq]`. In the future this will be interpreted as an array index, `arr[np.array(seq)]`, which will result either in an error or a different result.\n",
" return np.add.reduce(sorted[indexer] * weights, axis=axis) / sumval\n"
]
},
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%matplotlib inline\n",
"sns.set(color_codes=True)\n",
"plt.xlim(30,90)\n",
"plt.ylim(0,1)\n",
"sns.regplot(x='Temperature', y='Frequency', data=data, logistic=True)\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**I think I have managed to correctly compute and plot the uncertainty of my prediction.** Although the shaded area seems very similar to [the one obtained by with R](https://app-learninglab.inria.fr/gitlab/moocrr-session1/moocrr-reproducibility-study/raw/5c9dbef11b4d7638b7ddf2ea71026e7bf00fcfb0/challenger.pdf), I can spot a few differences (e.g., the blue point for temperature 63 is outside)... Could this be a numerical error ? Or a difference in the statistical method ? It is not clear which one is \"right\"."
]
}
],
"metadata": {
"celltoolbar": "Hide code",
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.7.0"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
src/Python3/challenger_Python_org.org 0 → 100644
View file @ 9d85e5f4
# -*- coding: utf-8 -*-
# -*- mode: org -*-
#+TITLE: Challenger - Python - Emacs - Windows 7 64 bits
* Risk Analysis of the Space Shuttle: Pre-Challenger Prediction of Failure
In this document we reperform some of the analysis provided in
/Risk Analysis of the Space Shuttle: Pre-Challenger Prediction of
Failure/ by /Siddhartha R. Dalal, Edward B. Fowlkes, Bruce Hoadley/
published in /Journal of the American Statistical Association/, Vol. 84,
No. 408 (Dec., 1989), pp. 945-957 and available at
http://www.jstor.org/stable/2290069.
On the fourth page of this article, they indicate that the maximum
likelihood estimates of the logistic regression using only temperature
are: *$\hat{\alpha}$ = 5.085* and *$\hat{\beta}$ = -0.1156* and their
asymptotic standard errors are *$s_{\hat{\alpha}}$ = 3.052* and
*$s_{\hat{\beta}}$ = 0.047*. The Goodness of fit indicated for this model was
*$G^2$ = 18.086* with *21* degrees of freedom. Our goal is to reproduce
the computation behind these values and the Figure 4 of this article,
possibly in a nicer looking way.
** Technical information on the computer on which the analysis is run
We will be using the Python 3 language using the pandas, statsmodels,
and numpy library.
#+begin_src python :results output :session :exports both
def print_imported_modules():
import sys
for name, val in sorted(sys.modules.items()):
if(hasattr(val, '__version__')):
print(val.__name__, val.__version__)
# else:
# print(val.__name__, "(unknown version)")
def print_sys_info():
import sys
import platform
print(sys.version)
print(platform.uname())
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import statsmodels.api as sm
import seaborn as sns
print_sys_info()
print_imported_modules()
#+end_src
** Loading and inspecting data
Let's start by reading data.
#+begin_src python :results output :session :exports both
data = pd.read_csv("https://app-learninglab.inria.fr/gitlab/moocrr-session1/moocrr-reproducibility-study/raw/master/data/shuttle.csv")
print(data)
#+end_src
We know from our previous experience on this data set that filtering
data is a really bad idea. We will therefore process it as such.
#+begin_src python :results file :session :var matplot_lib_filename=(org-babel-temp-file "figure" ".png") :exports both
%matplotlib inline
pd.set_option('mode.chained_assignment',None) # this removes a useless warning from pandas
data["Frequency"]=data.Malfunction/data.Count
data.plot(x="Temperature",y="Frequency",kind="scatter",ylim=[0,1])
plt.grid(True)
plt.tight_layout()
plt.savefig(matplot_lib_filename)
matplot_lib_filename
#+end_src
** Logistic regression
Let's assume O-rings independently fail with the same probability
which solely depends on temperature. A logistic regression should
allow us to estimate the influence of temperature.
#+begin_src python :results output :session :exports both
import statsmodels.api as sm
data["Success"]=data.Count-data.Malfunction
data["Intercept"]=1
logmodel=sm.GLM(data['Frequency'], data[['Intercept','Temperature']],
family=sm.families.Binomial(sm.families.links.logit)).fit()
print(logmodel.summary())
#+end_src
The maximum likelyhood estimator of the intercept and of Temperature
are thus *$\hat{\alpha}$ = 5.0850* and *$\hat{\beta}$ = -0.1156*. This *corresponds*
to the values from the article of Dalal /et al./ The standard errors are
/$s_{\hat{\alpha}}$ = 7.477/ and /$s_{\hat{\beta}}$ = 0.115/, which is *different* from
the *3.052* and *0.04702* reported by Dallal /et al./ The deviance is
/3.01444/ with *21* degrees of freedom. I cannot find any value similar
to the Goodness of fit (*$G^2$ = 18.086*) reported by Dalal /et al./ There
seems to be something wrong. Oh I know, I haven't indicated that my
observations are actually the result of 6 observations for each rocket
launch. Let's indicate these weights (since the weights are always the
same throughout all experiments, it does not change the estimates of
the fit but it does influence the variance estimates).
#+begin_src python :results output :session :exports both
logmodel=sm.GLM(data['Frequency'], data[['Intercept','Temperature']],
family=sm.families.Binomial(sm.families.links.logit),
var_weights=data['Count']).fit()
print(logmodel.summary())
#+end_src
Good, now I have recovered the asymptotic standard errors
*$s_{\hat{\alpha}}$ = 3.052* and *$s_{\hat{\beta}}$ = 0.047*. The Goodness of fit
(Deviance) indicated for this model is *$G^2$ = 18.086* with *21* degrees
of freedom (Df Residuals).
*I have therefore managed to fully replicate the results of the Dalal
/et al./ article*.
** Predicting failure probability
The temperature when launching the shuttle was 31°F. Let's try to
estimate the failure probability for such temperature using our model:
#+begin_src python :results output :session :exports both
data_pred = pd.DataFrame({'Temperature': np.linspace(start=30, stop=90, num=121),
'Intercept': 1})
data_pred['Frequency'] = logmodel.predict(data_pred)
print(data_pred.head())
#+end_src
#+begin_src python :results file :session :var matplot_lib_filename=(org-babel-temp-file "figure" ".png") :exports both
%matplotlib inline
data_pred.plot(x="Temperature",y="Frequency",kind="line",ylim=[0,1])
plt.scatter(x=data["Temperature"],y=data["Frequency"])
plt.grid(True)
plt.savefig(matplot_lib_filename)
matplot_lib_filename
#+end_src
La fonction =logmodel.predict(data_pred)= ne fonctionne pas avec les
dernières versions de pandas (Frequency = 1 pour toutes les températures).
On peut alors utiliser le code suivant pour calculer les prédictions
et tracer la courbe :
#+begin_src python :results output :session :exports both
# Inspiring from http://blog.yhat.com/posts/logistic-regression-and-python.html
def logit_inv(x):
return(np.exp(x)/(np.exp(x)+1))
data_pred['Prob']=logit_inv(data_pred['Temperature'] * logmodel.params['Temperature'] +
logmodel.params['Intercept'])
print(data_pred.head())
#+end_src
#+begin_src python :results file :session :var matplot_lib_filename=(org-babel-temp-file "figure" ".png") :exports both
%matplotlib inline
data_pred.plot(x="Temperature",y="Prob",kind="line",ylim=[0,1])
plt.scatter(x=data["Temperature"],y=data["Frequency"])
plt.grid(True)
plt.savefig(matplot_lib_filename)
matplot_lib_filename
#+end_src
This figure is very similar to the Figure 4 of Dalal /et al./ *I have
managed to replicate the Figure 4 of the Dalal /et al./ article.*
** Computing and plotting uncertainty
Following the documentation of
[Seaborn](https://seaborn.pydata.org/generated/seaborn.regplot.html),
I use regplot.
#+begin_src python :results file :session :var matplot_lib_filename=(org-babel-temp-file "figure" ".png") :exports both
sns.set(color_codes=True)
plt.xlim(30,90)
plt.ylim(0,1)
sns.regplot(x='Temperature', y='Frequency', data=data, logistic=True)
plt.show()
plt.savefig(matplot_lib_filename)
matplot_lib_filename
#+end_src
**I think I have managed to correctly compute and plot the uncertainty
of my prediction.** Although the shaded area seems very similar to
[the one obtained by with
R](https://app-learninglab.inria.fr/gitlab/moocrr-session1/moocrr-reproducibility-study/raw/5c9dbef11b4d7638b7ddf2ea71026e7bf00fcfb0/challenger.pdf),
I can spot a few differences (e.g., the blue point for temperature
63 is outside)... Could this be a numerical error ? Or a difference
in the statistical method ? It is not clear which one is "right".
src/R/Challenger.html 0 → 100644
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---
title: "Risk Analysis of the Space Shuttle: Pre-Challenger Prediction of Failure"
author: "Arnaud Legrand"
date: "25 October 2018"
output: pdf_document
---
In this document we reperform some of the analysis provided in
*Risk Analysis of the Space Shuttle: Pre-Challenger Prediction of Failure* by *Siddhartha R. Dalal, Edward B. Fowlkes, Bruce Hoadley* published in *Journal of the American Statistical Association*, Vol. 84, No. 408 (Dec., 1989), pp. 945-957 and available at http://www.jstor.org/stable/2290069.
On the fourth page of this article, they indicate that the maximum likelihood estimates of the logistic regression using only temperature are: $\hat{\alpha}=5.085$ and $\hat{\beta}=-0.1156$ and their asymptotic standard errors are $s_{\hat{\alpha}}=3.052$ and $s_{\hat{\beta}}=0.047$. The Goodness of fit indicated for this model was $G^2=18.086$ with 21 degrees of freedom. Our goal is to reproduce the computation behind these values and the Figure 4 of this article, possibly in a nicer looking way.
# Technical information on the computer on which the analysis is run
We will be using the R language using the ggplot2 library.
```{r}
library(ggplot2)
sessionInfo()
```
Here are the available libraries
```{r}
devtools::session_info()
```
# Loading and inspecting data
Let's start by reading data:
```{r}
data = read.csv("https://app-learninglab.inria.fr/gitlab/moocrr-session1/moocrr-reproducibility-study/raw/master/data/shuttle.csv",header=T)
data
```
We know from our previous experience on this data set that filtering data is a really bad idea. We will therefore process it as such.
Let's visually inspect how temperature affects malfunction:
```{r}
plot(data=data, Malfunction/Count ~ Temperature, ylim=c(0,1))
```
# Logistic regression
Let's assume O-rings independently fail with the same probability which solely depends on temperature. A logistic regression should allow us to estimate the influence of temperature.
```{r}
logistic_reg = glm(data=data, Malfunction/Count ~ Temperature, weights=Count,
family=binomial(link='logit'))
summary(logistic_reg)
```
The maximum likelyhood estimator of the intercept and of Temperature are thus $\hat{\alpha}=5.0849$ and $\hat{\beta}=-0.1156$ and their standard errors are $s_{\hat{\alpha}} = 3.052$ and $s_{\hat{\beta}} = 0.04702$. The Residual deviance corresponds to the Goodness of fit $G^2=18.086$ with 21 degrees of freedom. **I have therefore managed to replicate the results of the Dalal *et al.* article**.
# Predicting failure probability
The temperature when launching the shuttle was 31°F. Let's try to
estimate the failure probability for such temperature using our model.:
```{r}
# shuttle=shuttle[shuttle$r!=0,]
tempv = seq(from=30, to=90, by = .5)
rmv <- predict(logistic_reg,list(Temperature=tempv),type="response")
plot(tempv,rmv,type="l",ylim=c(0,1))
points(data=data, Malfunction/Count ~ Temperature)
```
This figure is very similar to the Figure 4 of Dalal et al. **I have managed to replicate the Figure 4 of the Dalal *et al.* article.**
# Confidence on the prediction
Let's try to plot confidence intervals with ggplot2.
```{r, fig.height=3.3}
ggplot(data, aes(y=Malfunction/Count, x=Temperature)) + geom_point(alpha=.2, size = 2, color="blue") +
geom_smooth(method = "glm", method.args = list(family = "binomial"), fullrange=T) +
xlim(30,90) + ylim(0,1) + theme_bw()
```
Mmmh, I have a warning from ggplot2 indicating *"non-integer #successes in a binomial glm!"*. This seems fishy. Furthermore, this confidence region seems huge... It seems strange to me that the uncertainty grows so large for higher temperatures. And compared to my previous call to glm, I haven't indicated the weight which accounts for the fact that each ratio Malfunction/Count corresponds to Count observations (if someone knows how to do this...). There must be something wrong.
So let's provide the "raw" data to ggplot2.
```{r}
data_flat=data.frame()
for(i in 1:nrow(data)) {
temperature = data[i,"Temperature"];
malfunction = data[i,"Malfunction"];
d = data.frame(Temperature=temperature,Malfunction=rep(0,times = data[i,"Count"]))
if(malfunction>0) {
d[1:malfunction, "Malfunction"]=1;
}
data_flat=rbind(data_flat,d)
}
dim(data_flat)
str(data_flat)
```
Let's check whether I obtain the same regression or not:
```{r}
logistic_reg_flat = glm(data=data_flat, Malfunction ~ Temperature, family=binomial(link='logit'))
summary(logistic_reg)
```
Perfect. The estimates and the standard errors are the same although the Residual deviance is difference since the distance is now measured with respect to each 0/1 measurement and not to ratios. Let's use plot the regression for *data_flat* along with the ratios (*data*).
```{r, fig.height=3.3}
ggplot(data=data_flat, aes(y=Malfunction, x=Temperature)) +
geom_smooth(method = "glm", method.args = list(family = "binomial"), fullrange=T) +
geom_point(data=data, aes(y=Malfunction/Count, x=Temperature),alpha=.2, size = 2, color="blue") +
geom_point(alpha=.5, size = .5) +
xlim(30,90) + ylim(0,1) + theme_bw()
```
This confidence interval seems much more reasonable (in accordance with the data) than the previous one. Let's check whether it corresponds to the prediction obtained when calling directly predict. Obtaining the prediction can be done directly or through the link function.
Here is the "direct" (response) version I used in my very first plot:
```{r}
pred = predict(logistic_reg_flat,list(Temperature=30),type="response",se.fit = T)
pred
```
The estimated Failure probability for 30° is thus $0.834$. However, the $se.fit$ value seems pretty hard to use as I can obviously not simply add $\pm 2 se.fit$ to $fit$ to compute a confidence interval.
Here is the "link" version:
```{r}
pred_link = predict(logistic_reg_flat,list(Temperature=30),type="link",se.fit = T)
pred_link
logistic_reg$family$linkinv(pred_link$fit)
```
I recover $0.834$ for the estimated Failure probability at 30°. But now, going through the *linkinv* function, we can use $se.fit$:
```{r}
critval = 1.96
logistic_reg$family$linkinv(c(pred_link$fit-critval*pred_link$se.fit,
pred_link$fit+critval*pred_link$se.fit))
```
The 95% confidence interval for our estimation is thus [0.163,0.992]. This is what ggplot2 just plotted me. This seems coherent.
**I am now rather confident that I have managed to correctly compute and plot the uncertainty of my prediction.** Let's be honnest, it took me a while. My first attempts were plainly wrong (I didn't know how to do this so I trusted ggplot2, which I was misusing) and did not use the correct statistical method. I also feel confident now because this has been somehow validated by other colleagues but it will be interesting that you collect other kind of plots values that you obtained, that differ and that you would probably have kept if you didn't have a reference to compare to. Please provide us with as many versions as you can.
src/R/challenger_R_org.org 0 → 100644
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# -*- coding: utf-8 -*-
# -*- mode: org -*-
#+TITLE: Risk Analysis of the Space Shuttle: Pre-Challenger Prediction of Failure - copied and edited version from Arnaud Legrand
#+AUTHOR: JFM
#+DATE: \today
In this document I reperform some of the analysis provided in /Risk Analysis of the Space Shuttle: Pre-Challenger Prediction of Failure/ by /Siddharta R. Dalal, Edward B. Fowlkes, Bruce Hoadley/ published in /Journal of the American Statistical Association/, Vol. 84, No. 408 (Dec., 1989), pp. 945-957 and available at [[http://www.jstor.org/stable/2290069]]. I use a version of this document proposed by Arnaud Legrand in the MOOC on reproducible research 2018. I want to see if I can replicate the analysis on my computer.
On the fourth page of this article, they indicate that the maximum likelihood estimates of the logistic regression using only temperature are: \hat{\alpha}=5.085 and \hat{\beta}=-0.1156 and their asymptotic standard errors are s_{\hat{\alpha}}=3.052 and s_{\hat{\beta}}=0.047. The Goodness of fit indicated for this model was G^2 =18.086 with 21 degrees of freedom. Our goal is to reproduce the computation behind these values and the Figure 4 of this article, possibly in a nicer looking way.
* Technical information on the computer on which the analysis is run
We will be using the R language using the ~ggplot2~ library.
#+begin_src R :results output :session *R* :exports both
library(ggplot2)
sessionInfo()
#+end_src
#+RESULTS:
#+begin_example
R version 3.5.1 (2018-07-02)
Platform: x86_64-suse-linux-gnu (64-bit)
Running under: openSUSE Leap 15.0
Matrix products: default
BLAS: /usr/lib64/R/lib/libRblas.so
LAPACK: /usr/lib64/R/lib/libRlapack.so
locale:
[1] LC_CTYPE=de_DE.UTF-8 LC_NUMERIC=C
[3] LC_TIME=de_DE.UTF-8 LC_COLLATE=de_DE.UTF-8
[5] LC_MONETARY=de_DE.UTF-8 LC_MESSAGES=de_DE.UTF-8
[7] LC_PAPER=de_DE.UTF-8 LC_NAME=C
[9] LC_ADDRESS=C LC_TELEPHONE=C
[11] LC_MEASUREMENT=de_DE.UTF-8 LC_IDENTIFICATION=C
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] ggplot2_3.0.0
loaded via a namespace (and not attached):
[1] colorspace_1.3-2 scales_1.0.0 compiler_3.5.1 plyr_1.8.4
[5] lazyeval_0.2.1 withr_2.1.2 pillar_1.3.0 gtable_0.2.0
[9] tibble_1.4.2 crayon_1.3.4 Rcpp_0.12.18 grid_3.5.1
[13] rlang_0.2.2 munsell_0.5.0
#+end_example
Here are the available libraries
#+begin_src R :results output :session *R* :exports both
devtools::session_info()
#+end_src
#+RESULTS:
#+begin_example
Session info-------------------------------------------------------------------
setting value
version R version 3.5.1 (2018-07-02)
system x86_64, linux-gnu
ui X11
language (EN)
collate de_DE.UTF-8
tz Europe/Berlin
Packages-----------------------------------------------------------------------
package * version date source
colorspace 1.3.2 2016-12-14 CRAN (R 3.5.1)
crayon 1.3.4 2017-09-16 CRAN (R 3.5.1)
devtools 1.6.1 2014-10-07 CRAN (R 3.5.1)
ggplot2 * 3.0.0 2018-07-03 CRAN (R 3.5.1)
gtable 0.2.0 2016-02-26 CRAN (R 3.5.1)
lazyeval 0.2.1 2017-10-29 CRAN (R 3.5.1)
munsell 0.5.0 2018-06-12 CRAN (R 3.5.1)
pillar 1.3.0 2018-07-14 CRAN (R 3.5.1)
plyr 1.8.4 2016-06-08 CRAN (R 3.5.1)
Rcpp 0.12.18 2018-07-23 CRAN (R 3.5.1)
rlang 0.2.2 2018-08-16 CRAN (R 3.5.1)
rstudioapi 0.7 2017-09-07 CRAN (R 3.5.1)
scales 1.0.0 2018-08-09 CRAN (R 3.5.1)
tibble 1.4.2 2018-01-22 CRAN (R 3.5.1)
withr 2.1.2 2018-03-15 CRAN (R 3.5.1)
#+end_example
* Loading and inspecting data
Let's start by reading data:
#+begin_src R :results output :session *R* :exports both
data = read.csv("https://app-learninglab.inria.fr/gitlab/moocrr-session1/moocrr-reproducibility-study/raw/master/data/shuttle.csv")
data
#+end_src
#+RESULTS:
#+begin_example
Date Count Temperature Pressure Malfunction
1 4/12/81 6 66 50 0
2 11/12/81 6 70 50 1
3 3/22/82 6 69 50 0
4 11/11/82 6 68 50 0
5 4/04/83 6 67 50 0
6 6/18/82 6 72 50 0
7 8/30/83 6 73 100 0
8 11/28/83 6 70 100 0
9 2/03/84 6 57 200 1
10 4/06/84 6 63 200 1
11 8/30/84 6 70 200 1
12 10/05/84 6 78 200 0
13 11/08/84 6 67 200 0
14 1/24/85 6 53 200 2
15 4/12/85 6 67 200 0
16 4/29/85 6 75 200 0
17 6/17/85 6 70 200 0
18 7/2903/85 6 81 200 0
19 8/27/85 6 76 200 0
20 10/03/85 6 79 200 0
21 10/30/85 6 75 200 2
22 11/26/85 6 76 200 0
23 1/12/86 6 58 200 1
#+end_example
We know from our previous experience on this data set that filtering data is a really bad idea. We ill therefore process it as such.
Let's visually inspect how temperature affects malfunction:
#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R*
plot(data=data, Malfunction/Count - Temperature, ylim = c(0,1))
#+end_src
#+RESULTS:
[[file:/tmp/babel-24411vWV/figure24411_ii.png]]
* Logistic regression
Let's assume O-rings independently fail with the same probability which solely depends on temperature. A logistic regression should allow us to estimate the influence of temperature.
#+begin_src R :results output :session *R* :exports both
logistic_reg = glm(data=data, Malfunction/Count - Temperature, weights=Count, family=binomial(link='logit'))
summary(logistic_reg)
#+end_src
#+RESULTS:
: Fehler in stats::model.frame(formula = Malfunction/Count - Temperature, :
: Objekt 'Malfunction' nicht gefunden
:
: Fehler in summary(logistic_reg) : Objekt 'logistic_reg' nicht gefunden
The maximum likelyhood estimatator of the intercept and of Temperature are thus \hat{\alpha}=? and \hat{\beta}=? and their standard errors are s_{\hat{\alpha}}=? and s_{\hat{\beta}}=?. The Residual deviance corresponds to the Goodness of fit G^2 = ? with ? degrees of freedom. Since some function does not operate as in the example given by Arnaud Legrand: *I have therefore _not yet_ managed to replicate the results of the Dalal /et.al./ article*.
* Predicting failure probability
The temperature when launching the shuttle was 31°F. Let's try to estimate the failure probability for such temperature using our model:
#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R*
# shuttle=shuttle[shuttle$r!=0,]
tempv = seq(from=30, to=90, by = .5)
rmv <- predict(logistic_reg,list(Temperature=tempv),type="response")
plot(tempv,rmv,type="1",ylim=c(0,1))
points(data=data, Malfunction/Count - Temperature)
#+end_src
#+RESULTS:
[[file:/tmp/babel-24411vWV/figure24411Z3u.png]]
For the error mentioned above I have not been able to plot this. This figure is _not yet_ very similar to the Figure 4 of Dalal et. al. *I have _not yet_ managed to replicate the Figure 4 of the Dalal /et. al./ article.*
* Confidence on the prediction
Let's try to plot confidence intervals with ~ggplot2~.
#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R*
ggplot(data, aes(y=Malfunction/Count, x=Temperature)) + geom_point(alpha=.2, size = 2, color="blue") + geom_smooth(method = "glm", method.args = list(family = "binomial"), fullrange=T) + xlim(30,90) + ylim(0,1) + theme_bw()
#+end_src
#+RESULTS:
[[file:/tmp/babel-24411vWV/figure24411mB1.png]]
Apparently I don't have the warning Arnaud Legrand mentions from ~ggplot2~ indicating /"non-integer #successes in a binomial glm!"/. This seems fishy for him, but not for me. But: yes, this confidence region seems huge... It seems strange to me that the uncertainty grows so large for higher temperatures. And compared to my previous call to ~glm~, I haven't indicated the weight which accounts for the fact that each ration Malfunction/Count corresponds to ~Count~ observations (if someone knows how to do this ...) There must be something wrong.
So let's provide the "raw" data to ~ggplot2~.
#+begin_src R :results output :session *R* :exports both
data_flat = data.frame()
for(i in 1:nrow(data)) {
temperature = data[i,"Temperature"];
malfunction = data[i,"Malfunction"];
d = data.frame(Temperature=temperature,Malfunction=rep(0,times = data[i,"Count"]))
if(malfunction>0) {
d[1:malfunction, "Malfunction"]=1;
}
data_flat=rbind(data_flat,d)
}
dim(data_flat)
#+end_src
#+RESULTS:
:
: [1] 138 2
#+begin_src R :results output :session *R* :exports both
str(data_flat)
#+end_src
#+RESULTS:
: 'data.frame': 138 obs. of 2 variables:
: $ Temperature: int 66 66 66 66 66 66 70 70 70 70 ...
: $ Malfunction: num 0 0 0 0 0 0 1 0 0 0 ...
Let's check whether I obtain the same regression or not:
#+begin_src R :results output :session *R* :exports both
logistic_reg_flat = glm(data=data_flat, Malfunction - Temperature, family=binomial(link='logit'))
summary(logistic_reg)
#+end_src
#+RESULTS:
: Fehler in stats::model.frame(formula = Malfunction - Temperature, data = data_flat, :
: Objekt 'Malfunction' nicht gefunden
:
: Fehler in summary(logistic_reg) : Objekt 'logistic_reg' nicht gefunden
So, again these objects are not here (same error as above, probably). So, for Arnaud Legrand this is perfect because he sees a result. For me it is not. The estimates and the standard errors for him are the same although the Residual deviance is difference since the distance is now measured with respect to each 0/1 measurement and not to rations. Let's use plot the regression for /~data_flat~/ along with the ratios (/~data~/).
#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R*
ggplot(data=data_flat, aes(y=Malfunction, x=Temperature)) + geom_smooth(method = "glm", method.args = list(family = "binomial"), fullrange=T) + geom_point(data=data, aes(y=Malfunction/Count, x=Temperature),alpha=.2, size = 2, color="blue") + geom_point(alpha=.5, size=.5) + xlim(30,90) + ylim(0,1) + theme_bw()
#+end_src
#+RESULTS:
[[file:/tmp/babel-24411vWV/figure24411YLE.png]]
This confidence interval seems much more reasonable (in accordance with the data) than the previous one. Let's check whether it corresponds to the prediction obtained when calling directly predict. Obtaining the prediction can be done directly or through the link function.
Here is the "direct" (response) version I used in my very first plot:
#+begin_src R :results output :session *R* :exports both
pred = predict(logistic_reg_flat,list(Temperature=30),type="response",se.fit = T)
pred
#+end_src
#+RESULTS:
: Fehler in predict(logistic_reg_flat, list(Temperature = 30), type = "response", :
: Objekt 'logistic_reg_flat' nicht gefunden
:
: Fehler: Objekt 'pred' nicht gefunden
Again, in my version of this document I cannot find the above defined object anymore. So, I cannot replicate what Arnaud Legrand wrote: The estimated Failure probability for 30° is thus ??. However the /se.fit/ value seems pretty hard to use as I can obviously not simply add \pm2 /se.fit/ to /fit/ to compute a confidence interval.
Here is the "link" version:
#+begin_src R :results output :session *R* :exports both
pred_link = predict(logistic_reg_flat,list(Temperature=39), type="link",se.fit = T)
pred.link
#+end_src
#+RESULTS:
: Fehler in predict(logistic_reg_flat, list(Temperature = 39), type = "link", :
: Objekt 'logistic_reg_flat' nicht gefunden
:
: Fehler: Objekt 'pred.link' nicht gefunden
#+begin_src R :results output :session *R* :exports both
logistic_reg$family$linkinv(pred_link$fit)
#+end_src
#+RESULTS:
: Fehler: Objekt 'logistic_reg' nicht gefunden
I recover ?? for the Estimated Failure probability at 30°. But now, going through the /linkinv/ function, we can use /se.fit/:
#+begin_src R :results output :session *R* :exports both
critval = 1.96
logistic_reg$family$linkinv(c(pred_link$fit-critval*pred_link$se.fit, pred_link$fit+critval*pred_link$se.fit))
#+end_src
#+RESULTS:
:
: Fehler: Objekt 'logistic_reg' nicht gefunden
The 95% confidence interval for our estimation is thus [??,??]. This is what ~ggplot2~ just plotted me. This seems coherent.
*I am now _not yet_ rather confident that I have managed to correctly compute and plot uncertainty of my prediction.* Let's be honnest, it took me a wile. My first attempts were plainly wrong (I didn't know how to do this so I trusted ~ggplot2~, which I was misusing) and did not use the correct statistical method. I also feel confident now becuase this has been somehow validated by other colleagues but it will be interesting that you collect other kind of plot values that you obtained ,that differ and that you would probably have kept if you didn't have a reference to compare to . Please, provide us with as many versions as you can.
So, I'm disappointed because some error in R or in my config leads to the fact that some objects are forgotten between blocks. I will try to export the whole document in pdf to see if that changes. Unfortunately, it doesn't. Right now I do not have the time to figure out by myself how to change this. So I will only upload this document and hope it still contributes to the database in some way.
src/R/challenger_R_org_Windows_64bits.org 0 → 100644
View file @ 9d85e5f4
# -*- coding: utf-8 -*-
# -*- mode: org -*-
#+TITLE: Challenger - R - Emacs - Windows 7 64 bits
#+LATEX_HEADER: \usepackage{parskip}
* Risk Analysis of the Space Shuttle: Pre-Challenger Prediction of Failure
In this document we reperform some of the analysis provided in
/Risk Analysis of the Space Shuttle: Pre-Challenger Prediction of
Failure/ by /Siddhartha R. Dalal, Edward B. Fowlkes, Bruce Hoadley/
published in /Journal of the American Statistical Association/, Vol. 84,
No. 408 (Dec., 1989), pp. 945-957 and available at
http://www.jstor.org/stable/2290069.
On the fourth page of this article, they indicate that the maximum
likelihood estimates of the logistic regression using only temperature
are: *$\hat{\alpha}$ = 5.085* and *$\hat{\beta}$ = -0.1156* and their
asymptotic standard errors are *$s_{\hat{\alpha}}$ = 3.052* and
*$s_{\hat{\beta}}$ = 0.047*. The Goodness of fit indicated for this model was
*$G^2$ = 18.086* with *21* degrees of freedom. Our goal is to reproduce
the computation behind these values and the Figure 4 of this article,
possibly in a nicer looking way.
* Technical information on the computer on which the analysis is run
We will be using the R language using the ~ggplot2~ library.
#+begin_src R :results output :session *R* :exports both
library(ggplot2)
sessionInfo()
#+end_src
#+RESULTS:
#+begin_example
R version 3.5.1 (2018-07-02)
Platform: i386-w64-mingw32/i386 (32-bit)
Running under: Windows 7 x64 (build 7601) Service Pack 1
Matrix products: default
locale:
[1] LC_COLLATE=French_France.1252 LC_CTYPE=French_France.1252
[3] LC_MONETARY=French_France.1252 LC_NUMERIC=C
[5] LC_TIME=French_France.1252
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] ggplot2_3.1.0
loaded via a namespace (and not attached):
[1] Rcpp_1.0.0 withr_2.1.2 crayon_1.3.4 dplyr_0.7.8
[5] assertthat_0.2.0 grid_3.5.1 plyr_1.8.4 R6_2.3.0
[9] gtable_0.2.0 magrittr_1.5 scales_1.0.0 pillar_1.3.0
[13] rlang_0.3.0.1 lazyeval_0.2.1 bindrcpp_0.2.2 glue_1.3.0
[17] purrr_0.2.5 munsell_0.5.0 compiler_3.5.1 pkgconfig_2.0.2
[21] colorspace_1.3-2 tidyselect_0.2.5 bindr_0.1.1 tibble_1.4.2
#+end_example
Here are the available libraries
#+begin_src R :results output :session *R* :exports both
devtools::session_info()
#+end_src
#+RESULTS:
#+begin_example
- Session info ---------------------------------------------------------------
setting value
version R version 3.5.1 (2018-07-02)
os Windows 7 x64 SP 1
system i386, mingw32
ui RTerm
language (EN)
collate French_France.1252
ctype French_France.1252
tz Europe/Paris
date 2018-12-10
- Packages -------------------------------------------------------------------
package * version date lib source
assertthat 0.2.0 2017-04-11 [1] CRAN (R 3.5.1)
backports 1.1.2 2017-12-13 [1] CRAN (R 3.5.0)
base64enc 0.1-3 2015-07-28 [1] CRAN (R 3.5.0)
bindr 0.1.1 2018-03-13 [1] CRAN (R 3.5.1)
bindrcpp 0.2.2 2018-03-29 [1] CRAN (R 3.5.1)
callr 3.0.0 2018-08-24 [1] CRAN (R 3.5.1)
cli 1.0.1 2018-09-25 [1] CRAN (R 3.5.1)
colorspace 1.3-2 2016-12-14 [1] CRAN (R 3.5.1)
crayon 1.3.4 2017-09-16 [1] CRAN (R 3.5.1)
desc 1.2.0 2018-05-01 [1] CRAN (R 3.5.1)
devtools 2.0.1 2018-10-26 [1] CRAN (R 3.5.1)
digest 0.6.18 2018-10-10 [1] CRAN (R 3.5.1)
dplyr 0.7.8 2018-11-10 [1] CRAN (R 3.5.1)
fs 1.2.6 2018-08-23 [1] CRAN (R 3.5.1)
ggplot2 * 3.1.0 2018-10-25 [1] CRAN (R 3.5.1)
glue 1.3.0 2018-07-17 [1] CRAN (R 3.5.1)
gtable 0.2.0 2016-02-26 [1] CRAN (R 3.5.1)
lazyeval 0.2.1 2017-10-29 [1] CRAN (R 3.5.1)
magrittr 1.5 2014-11-22 [1] CRAN (R 3.5.1)
memoise 1.1.0 2017-04-21 [1] CRAN (R 3.5.1)
munsell 0.5.0 2018-06-12 [1] CRAN (R 3.5.1)
pillar 1.3.0 2018-07-14 [1] CRAN (R 3.5.1)
pkgbuild 1.0.2 2018-10-16 [1] CRAN (R 3.5.1)
pkgconfig 2.0.2 2018-08-16 [1] CRAN (R 3.5.1)
pkgload 1.0.2 2018-10-29 [1] CRAN (R 3.5.1)
plyr 1.8.4 2016-06-08 [1] CRAN (R 3.5.1)
prettyunits 1.0.2 2015-07-13 [1] CRAN (R 3.5.1)
processx 3.2.0 2018-08-16 [1] CRAN (R 3.5.1)
ps 1.2.1 2018-11-06 [1] CRAN (R 3.5.1)
purrr 0.2.5 2018-05-29 [1] CRAN (R 3.5.1)
R6 2.3.0 2018-10-04 [1] CRAN (R 3.5.1)
Rcpp 1.0.0 2018-11-07 [1] CRAN (R 3.5.1)
remotes 2.0.2 2018-10-30 [1] CRAN (R 3.5.1)
rlang 0.3.0.1 2018-10-25 [1] CRAN (R 3.5.1)
rprojroot 1.3-2 2018-01-03 [1] CRAN (R 3.5.1)
scales 1.0.0 2018-08-09 [1] CRAN (R 3.5.1)
sessioninfo 1.1.1 2018-11-05 [1] CRAN (R 3.5.1)
testthat 2.0.1 2018-10-13 [1] CRAN (R 3.5.1)
tibble 1.4.2 2018-01-22 [1] CRAN (R 3.5.1)
tidyselect 0.2.5 2018-10-11 [1] CRAN (R 3.5.1)
usethis 1.4.0 2018-08-14 [1] CRAN (R 3.5.1)
withr 2.1.2 2018-03-15 [1] CRAN (R 3.5.1)
[1] c:/Program Files/R/R-3.5.1/library
#+end_example
* Loading and inspecting data
Let's start by reading data:
#+begin_src R :results output :session *R* :exports both
data = read.csv("https://app-learninglab.inria.fr/gitlab/moocrr-session1/moocrr-reproducibility-study/raw/master/data/shuttle.csv")
data
#+end_src
#+RESULTS:
#+begin_example
Date Count Temperature Pressure Malfunction
1 4/12/81 6 66 50 0
2 11/12/81 6 70 50 1
3 3/22/82 6 69 50 0
4 11/11/82 6 68 50 0
5 4/04/83 6 67 50 0
6 6/18/82 6 72 50 0
7 8/30/83 6 73 100 0
8 11/28/83 6 70 100 0
9 2/03/84 6 57 200 1
10 4/06/84 6 63 200 1
11 8/30/84 6 70 200 1
12 10/05/84 6 78 200 0
13 11/08/84 6 67 200 0
14 1/24/85 6 53 200 2
15 4/12/85 6 67 200 0
16 4/29/85 6 75 200 0
17 6/17/85 6 70 200 0
18 7/2903/85 6 81 200 0
19 8/27/85 6 76 200 0
20 10/03/85 6 79 200 0
21 10/30/85 6 75 200 2
22 11/26/85 6 76 200 0
23 1/12/86 6 58 200 1
#+end_example
We know from our previous experience on this data set that filtering
data is a really bad idea. We will therefore process it as such.
Let's visually inspect how temperature affects malfunction:
#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R*
plot(data=data, Malfunction/Count ~ Temperature, ylim = c(0,1))
#+end_src
* Logistic regression
Let's assume O-rings independently fail with the same probability
which solely depends on temperature. A logistic regression should
allow us to estimate the influence of temperature.
#+begin_src R :results output :session *R* :exports both
logistic_reg = glm(data=data, Malfunction/Count ~ Temperature, weights=Count,
family=binomial(link='logit'))
summary(logistic_reg)
#+end_src
#+RESULTS:
#+begin_example
Call:
glm(formula = Malfunction/Count ~ Temperature, family = binomial(link = "logit"),
data = data, weights = Count)
Deviance Residuals:
Min 1Q Median 3Q Max
-0.95227 -0.78299 -0.54117 -0.04379 2.65152
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 5.08498 3.05247 1.666 0.0957 .
Temperature -0.11560 0.04702 -2.458 0.0140 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 24.230 on 22 degrees of freedom
Residual deviance: 18.086 on 21 degrees of freedom
AIC: 35.647
Number of Fisher Scoring iterations: 5
#+end_example
The maximum likelyhood estimatator of the intercept and of Temperature
are thus *$\hat{\alpha}$ = 5.0850* and *$\hat{\beta}$ = -0.1156* and their standard
errors are *$s_{\hat{\alpha}}$ = 3.052* and *$s_{\hat{\beta}}$ = 0.04702*. The Residual
deviance corresponds to the Goodness of fit *$G^2$ = 18.086* with *21*
degrees of freedom. *I have therefore managed to replicate the results
of the Dalal /et al./ article*.
* Predicting failure probability
The temperature when launching the shuttle was 31°F. Let's try to
estimate the failure probability for such temperature using our model:
#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R*
# shuttle=shuttle[shuttle$r!=0,]
tempv = seq(from=30, to=90, by = .5)
rmv <- predict(logistic_reg,list(Temperature=tempv),type="response")
plot(tempv,rmv,type="l",ylim=c(0,1))
points(data=data, Malfunction/Count ~ Temperature)
#+end_src
This figure is very similar to the Figure 4 of Dalal /et al./ *I have
managed to replicate the Figure 4 of the Dalal /et al./ article.*
* Confidence on the prediction
Let's try to plot confidence intervals with ~ggplot2~.
#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R*
ggplot(data, aes(y=Malfunction/Count, x=Temperature)) +
geom_point(alpha=.2, size = 2, color="blue") +
geom_smooth(method = "glm", method.args = list(family = "binomial"),
fullrange=T) +
xlim(30,90) + ylim(0,1) + theme_bw()
#+end_src
I don't get any warning from ~ggplot2~ indicating /"non-integer
#successes in a binomial glm!"/ but this confidence region seems
huge... It seems strange to me that the uncertainty grows so large for
higher temperatures. And compared to my previous call to ~glm~, I
haven't indicated the weight which accounts for the fact that each
ration Malfunction/Count corresponds to ~Count~ observations (if someone
knows how to do this...). There must be something wrong.
So let's provide the "raw" data to ~ggplot2~.
#+begin_src R :results output :session *R* :exports both
data_flat = data.frame()
for(i in 1:nrow(data)) {
temperature = data[i,"Temperature"];
malfunction = data[i,"Malfunction"];
d = data.frame(Temperature=temperature,Malfunction=rep(0,times = data[i,"Count"]))
if(malfunction>0) {
d[1:malfunction, "Malfunction"]=1;
}
data_flat=rbind(data_flat,d)
}
dim(data_flat)
#+end_src
#+RESULTS:
: [1] 138 2
#+begin_src R :results output :session *R* :exports both
str(data_flat)
#+end_src
#+RESULTS:
: 'data.frame': 138 obs. of 2 variables:
: $ Temperature: int 66 66 66 66 66 66 70 70 70 70 ...
: $ Malfunction: num 0 0 0 0 0 0 1 0 0 0 ...
Let's check whether I obtain the same regression or not:
#+begin_src R :results output :session *R* :exports both
logistic_reg_flat = glm(data=data_flat, Malfunction ~ Temperature,
family=binomial(link='logit'))
summary(logistic_reg)
#+end_src
#+RESULTS:
#+begin_example
Call:
glm(formula = Malfunction/Count ~ Temperature, family = binomial(link = "logit"),
data = data, weights = Count)
Deviance Residuals:
Min 1Q Median 3Q Max
-0.95227 -0.78299 -0.54117 -0.04379 2.65152
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 5.08498 3.05247 1.666 0.0957 .
Temperature -0.11560 0.04702 -2.458 0.0140 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 24.230 on 22 degrees of freedom
Residual deviance: 18.086 on 21 degrees of freedom
AIC: 35.647
Number of Fisher Scoring iterations: 5
#+end_example
Perfect. The estimates and the standard errors for him are the same
although the Residual deviance is difference since the distance is now
measured with respect to each =0/1= measurement and not to
rations. Let's use plot the regression for /data_flat/ along with the
ratios (/data/).
#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R*
ggplot(data=data_flat, aes(y=Malfunction, x=Temperature)) +
geom_smooth(method = "glm", method.args = list(family = "binomial"),
fullrange=T) +
geom_point(data=data, aes(y=Malfunction/Count, x=Temperature),
alpha=.2, size = 2, color="blue") +
geom_point(alpha=.5, size=.5) + xlim(30,90) + ylim(0,1) + theme_bw()
#+end_src
This confidence interval seems much more reasonable (in accordance
with the data) than the previous one. Let's check whether it
corresponds to the prediction obtained when calling directly
predict. Obtaining the prediction can be done directly or through the
link function.
Here is the "direct" (response) version I used in my very first plot:
#+begin_src R :results output :session *R* :exports both
pred = predict(logistic_reg_flat, list(Temperature=30), type="response", se.fit = T)
pred
#+end_src
#+RESULTS:
#+begin_example
$fit
1
0.834373
$se.fit
1
0.2293304
$residual.scale
[1] 1
#+end_example
The estimated Failure probability for 30° is thus 0.834. However the
/se.fit/ value seems pretty hard to use as I can obviously not simply
add \pm2 /se.fit/ to /fit/ to compute a confidence interval.
Here is the "link" version:
#+begin_src R :results output :session *R* :exports both
pred_link = predict(logistic_reg_flat, list(Temperature=30), type="link", se.fit = T)
pred_link
#+end_src
#+RESULTS:
: $fit
: 1
: 1.616942
:
: $se.fit
: [1] 1.659473
:
: $residual.scale
: [1] 1
#+begin_src R :results output :session *R* :exports both
logistic_reg$family$linkinv(pred_link$fit)
#+end_src
#+RESULTS:
: 1
: 0.834373
I recover 0.834 for the Estimated Failure probability at 30°. But now, going through the /linkinv/ function, we can use /se.fit/:
#+begin_src R :results output :session *R* :exports both
critval = 1.96
logistic_reg$family$linkinv(c(pred_link$fit-critval*pred_link$se.fit, pred_link$fit+critval*pred_link$se.fit))
#+end_src
#+RESULTS:
: 1 1
: 0.1630612 0.9923814
The 95% confidence interval for our estimation is thus [0.163,0.992]. This
is what ~ggplot2~ just plotted me. This seems coherent.
*I am now rather confident that I have managed to correctly compute and
plot uncertainty of my prediction.* Let's be honnest, it took me a
wile. My first attempts were plainly wrong (I didn't know how to do
this so I trusted ~ggplot2~, which I was misusing) and did not use the
correct statistical method. I also feel confident now becuase this has
been somehow validated by other colleagues but it will be interesting
that you collect other kind of plot values that you obtained, that
differ and that you would probably have kept if you didn't have a
reference to compare to. Please, provide us with as many versions as
you can.
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