"Les données de l'incidence du syndrome grippal sont disponibles du site Web du Réseau Sentinelles. Nous les récupérons sous forme d'un fichier en format CSV dont chaque ligne correspond à une semaine de la période demandée. Nous téléchargeons toujours le jeu de données complet, qui commence en 1984 et se termine avec une semaine récente."
"# 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, Mar 13 2018, 01:15:57) \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",
"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",
"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": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"/opt/conda/lib/python3.6/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",
"**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/moocrr/gitlab/moocrr-session3/moocrr-reproducibility-study/tree/master/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\"."
