Update exo5_en.Rmd

module2/exo5/exo5_en.Rmd

 ---
 title: "Analysis of the risk of failure of the O-rings on the Challenger shuttle"
-author: "Arnaud Legrand"
-date: "28 juin 2018"
+author: "Gkiouzepi Eleni"
+date: "24/7/2021"
 output: html_document
 ---
 
@@ -36,13 +36,12 @@ temperature (in Fahrenheit) and pressure (in psi), and finally the
 number of identified malfunctions.
 
 # Graphical inspection
-Flights without incidents do not provide any information
+~~Flights without incidents do not provide any information
 on the influence of temperature or pressure on malfunction.
-We thus focus on the experiments in which at least one O-ring was defective.
+We thus focus on the experiments in which at least one O-ring was defective.~~ **Wrong assumption**
 
 ```{r}
-data = data[data$Malfunction>0,]
-data
+# mal = data[data$Malfunction>0,]
 ```
 
 We have a high temperature variability but
@@ -71,49 +70,64 @@ logistic regression.
 logistic_reg = glm(data=data, Malfunction/Count ~ Temperature, weights=Count, 
                    family=binomial(link='logit'))
 summary(logistic_reg)
+
+# mal_logistic_reg = glm(data=mal, Malfunction/Count ~ Temperature, weights=Count, 
+#                    family=binomial(link='logit'))
+# summary(mal_logistic_reg)
 ```
 
-The most likely estimator of the temperature parameter is 0.001416
-and the standard error of this estimator is 0.049, in other words we
-cannot distinguish any particular impact and we must take our
+The most likely estimator of the temperature parameter is  ~~0.001416~~ __-0.11560__
+and the standard error of this estimator is 0.047, in other words
+**WRONG** ~~we
+cannot distinguish any particular impact~~
+_it is inverse-dependent on temperature, if temperature decreases by 1 degree, the probability of O-ring malfunction increases by 0.1156,_  and we must take our
 estimates with caution.
 
+
 # Estimation of the probability of O-ring malfunction
 The expected temperature on the take-off day is 31°F. Let's try to
 estimate the probability of O-ring malfunction at
 this temperature from the model we just built:
 
 ```{r}
-# shuttle=shuttle[shuttle$r!=0,] 
+# shuttle=shuttle[shuttle$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))
+# rmv_mal <- predict(mal_logistic_reg,list(Temperature=tempv),type="response")
+# plot(tempv,rmv_mal,type="l",ylim=c(0,1))
+# points(data=mal, Malfunction/Count ~ Temperature)
+
+
+rmv <- predict(logistic_reg,list(Temperature=tempv),se.fit=T,type="response")
+plot(tempv,rmv$fit,type="l",ylim=c(0,1))
+lines(tempv,rmv$fit+rmv$se.fit,col="red")
+lines(tempv,rmv$fit-rmv$se.fit,col="red")
 points(data=data, Malfunction/Count ~ Temperature)
 ```
 
-As expected from the initial data, the
-temperature has no significant impact on the probability of failure of the
-O-rings. It will be about 0.2, as in the tests
-where we had a failure of at least one joint. Let's get back to the initial dataset to estimate the probability of failure:
+~~As expected from the initial data~~, the
+temperature has **VERY** ~~no~~ significant impact on the probability of failure of the
+O-rings. It will be ~~about 0.2~~ **in average over 0.8 to as high as more than 1.0 (certain)**,~~as in the tests
+where we had a failure of at least one joint~~ **so we are expecting a failure of at least 4 joints**. Let's ~~get back to the initial dataset to~~ estimate the probability of failure:
 
 ```{r}
-data_full = read.csv("shuttle.csv",header=T)
-sum(data_full$Malfunction)/sum(data_full$Count)
+# data_full = read.csv("shuttle.csv",header=T)
+# sum(data_full$Malfunction)/sum(data_full$Count)
+
+estim = predict(logistic_reg,list(Temperature=31),se.fit=T,type="response")
+estim
 ```
 
-This probability is thus about $p=0.065$. Knowing that there is
+This probability is thus about $p=`r round(estim$fit, digits = 5)`\pm`r round(estim$se.fit, digits = 5)`$. Knowing that there is
 a primary and a secondary O-ring on each of the three parts of the
 launcher, the probability of failure of both joints of a launcher
-is $p^2 \approx 0.00425$. The probability of failure of any one of the
-launchers is $1-(1-p^2)^3 \approx 1.2%$.  That would really be
+is $p^2 \approx `r round((estim$fit+estim$se.fit)^2, digits = 2)`\pm`r round(2*estim$se.fit*estim$fit, digits = 2)`$. The probability of failure of any one of the
+launchers is $1-(1-p^2)^3 \approx `r (1-(1-round((estim$fit+estim$se.fit)^2, digits = 0))^3)*100`\%$.  ~~That would really be
 bad luck.... Everything is under control, so the takeoff can happen
-tomorrow as planned.
+tomorrow as planned~~.**ABORT! ABORT! ABORT THE MISSION!**
 
-But the next day, the Challenger shuttle exploded and took away
+*Unfortunately, none of the above analysis was carried out properly and* the next day, the Challenger shuttle exploded and took away
 with her the seven crew members. The public was shocked and in
 the subsequent investigation, the reliability of the
 O-rings was questioned. Beyond the internal communication problems
 of NASA, which have a lot to do with this fiasco, the previous analysis
-includes (at least) a small problem.... Can you find it?
-You are free to modify this analysis and to look at this dataset
-from all angles in order to to explain what's wrong.
+includes (at least) a small problem.