# -*- 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/moocrr/gitlab/moocrr-session3/moocrr-reproducibility-study/tree/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.
