if(!("sageR" %in% installed.packages())){install.packages("sageR")}
library(sageR)

## Intervalles de confiance

### Pour la moyenne

data(Flux)
mean(Flux)
#> [1] 13.99533
shapiro.test(Flux)
#>
#>  Shapiro-Wilk normality test
#>
#> data:  Flux
#> W = 0.97931, p-value = 0.8068
qqnorm(Flux)
qqline(Flux)

t.test(Flux)
#>
#>  One Sample t-test
#>
#> data:  Flux
#> t = 80.878, df = 29, p-value < 2.2e-16
#> alternative hypothesis: true mean is not equal to 0
#> 95 percent confidence interval:
#>  13.64142 14.34925
#> sample estimates:
#> mean of x
#>  13.99533

### Pour la variance

var(Flux)
#> [1] 0.8983154

L’hypothèse de normalité a déjà été validée au seuil $$\alpha=5\%$$

if(!("TeachingDemos" %in% installed.packages())){install.packages("TeachingDemos")}
library(TeachingDemos)
TeachingDemos::sigma.test(Flux)
#>
#>  One sample Chi-squared test for variance
#>
#> data:  Flux
#> X-squared = 26.051, df = 29, p-value = 0.7545
#> alternative hypothesis: true variance is not equal to 1
#> 95 percent confidence interval:
#>  0.5697691 1.6234206
#> sample estimates:
#> var of Flux
#>   0.8983154

## Test $$t$$ pour un échantillon

data(Precipitations_USA)
colnames(Precipitations_USA) <- c("Ville", "Precipitation (inches)", "Precipitation (cms)", "Etat")
if(!("ggmap" %in% installed.packages())){install.packages("ggmap")}
library(ggmap)
#> Please cite ggmap if you use it! See citation("ggmap") for details.
if(!("tidygeocoder" %in% installed.packages())){install.packages("tidygeocoder")}
library(tidygeocoder)
#>
#> Attaching package: 'tidygeocoder'
#> The following object is masked from 'package:ggmap':
#>
#>     geocode
levels(Precipitations_USA[,1]) <- gsub("Washington. D.C.","Washington",levels(Precipitations_USA[,1]))
gps_coords <- mapply(tidygeocoder::geo, city= Precipitations_USA[,1], state = Precipitations_USA[,4], method = "osm")
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gps_coords[1,gps_coords[1,]=="Washington"] <- "Washington. D.C."

gps_coords_df <- as.data.frame(matrix(unlist(gps_coords),ncol=4,byrow=TRUE))
colnames(gps_coords_df) <- c("Ville", "Etat", "lat", "long")
gps_coords_df$lat <- as.numeric(gps_coords_df$lat)
gps_coords_df$long <- as.numeric(gps_coords_df$long)

precip_gps <- merge(Precipitations_USA,gps_coords_df)
precip_gps_map <- ggmap::get_map(c(left = min(gps_coords_df$long)*1.03, bottom = min(gps_coords_df$lat)/1.2, right = max(gps_coords_df$long)/1.12, top = max(gps_coords_df$lat))*1.03, source = "osm", zoom=6, messaging = FALSE)
#> 209 tiles needed, this may take a while (try a smaller zoom).
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ggmap::ggmap(precip_gps_map,extent = "device") +
geom_point(aes(x = long, y = lat, color=Precipitation (cms)),
data = precip_gps, size = 5, pch = 20) +
scale_color_gradient(low="red", high="blue")

precip_gps_map_bw <- ggmap::get_stamenmap(c(left = min(gps_coords_df$long)*1.03, bottom = min(gps_coords_df$lat)/1.2, right = max(gps_coords_df$long)/1.12, top = max(gps_coords_df$lat))*1.03, source = "stamen", zoom=6, maptype="toner")
#> 209 tiles needed, this may take a while (try a smaller zoom).
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ggmap::ggmap(precip_gps_map_bw,extent = "device") +
geom_point(aes(x = long, y = lat, color=Precipitation (cms)),
data = precip_gps, size = 5, pch = 20, fill = "white") +
scale_color_gradient(low="white", high="black")

with(Precipitations_USA, shapiro.test(Precipitation (cms)))
#>
#>  Shapiro-Wilk normality test
#>
#> data:  Precipitation (cms)
#> W = 0.97241, p-value = 0.1911
with(Precipitations_USA,qqnorm(Precipitation (cms)))
with(Precipitations_USA,qqline(Precipitation (cms)))

with(Precipitations_USA, t.test(Precipitation (cms), mu = 83.90))
#>
#>  One Sample t-test
#>
#> data:  Precipitation (cms)
#> t = 1.6898, df = 59, p-value = 0.09635
#> alternative hypothesis: true mean is not equal to 83.9
#> 95 percent confidence interval:
#>  82.61107 99.18519
#> sample estimates:
#> mean of x
#>  90.89813
with(Precipitations_USA, TeachingDemos::sigma.test(Precipitation (cms)))
#>
#>  One sample Chi-squared test for variance
#>
#> data:  Precipitation (cms)
#> X-squared = 60717, df = 59, p-value < 2.2e-16
#> alternative hypothesis: true variance is not equal to 1
#> 95 percent confidence interval:
#>   739.3954 1530.8720
#> sample estimates:
#> var of Precipitation (cms)
#>                   1029.106
data(Resistance)
qqnorm(Resistance)
qqline(Resistance)

if(!("TeachingDemos" %in% installed.packages())){install.packages("TeachingDemos")}
TeachingDemos::sigma.test(Resistance,sigmasq = 8)
#>
#>  One sample Chi-squared test for variance
#>
#> data:  Resistance
#> X-squared = 38.47, df = 29, p-value = 0.2245
#> alternative hypothesis: true variance is not equal to 8
#> 95 percent confidence interval:
#>   6.731113 19.178694
#> sample estimates:
#> var of Resistance
#>          10.61248
binom.test(30,200,p=0.10,alternative=c("greater"))
#>
#>  Exact binomial test
#>
#> data:  30 and 200
#> number of successes = 30, number of trials = 200, p-value = 0.01633
#> alternative hypothesis: true probability of success is greater than 0.1
#> 95 percent confidence interval:
#>  0.1100814 1.0000000
#> sample estimates:
#> probability of success
#>                   0.15

## Test $$t$$ pour deux échantillons indépendants.

data(Essence)
shapiro.test(Essence$Aube) #> #> Shapiro-Wilk normality test #> #> data: Essence$Aube
#> W = 0.97524, p-value = 0.6897
with(Essence,qqnorm(Aube))
with(Essence,qqline(Aube))

shapiro.test(Essence$Marne) #> #> Shapiro-Wilk normality test #> #> data: Essence$Marne
#> W = 0.95879, p-value = 0.2884
with(Essence,qqnorm(Marne))
with(Essence,qqline(Marne))

mean(Essence$Aube) #> [1] 1.529867 mean(Essence$Marne)
#> [1] 1.4241
with(Essence, var.test(x=Aube,y=Marne))
#>
#>  F test to compare two variances
#>
#> data:  Aube and Marne
#> F = 1.6155, num df = 29, denom df = 29, p-value = 0.2025
#> alternative hypothesis: true ratio of variances is not equal to 1
#> 95 percent confidence interval:
#>  0.7689431 3.3942558
#> sample estimates:
#> ratio of variances
#>           1.615546
with(Essence, t.test(x=Aube,y=Marne, var.equal = TRUE))
#>
#>  Two Sample t-test
#>
#> data:  Aube and Marne
#> t = 21.852, df = 58, p-value < 2.2e-16
#> alternative hypothesis: true difference in means is not equal to 0
#> 95 percent confidence interval:
#>  0.0960779 0.1154554
#> sample estimates:
#> mean of x mean of y
#>  1.529867  1.424100
with(Essence, t.test(x=Aube,y=Marne))
#>
#>  Welch Two Sample t-test
#>
#> data:  Aube and Marne
#> t = 21.852, df = 54.956, p-value < 2.2e-16
#> alternative hypothesis: true difference in means is not equal to 0
#> 95 percent confidence interval:
#>  0.09606647 0.11546687
#> sample estimates:
#> mean of x mean of y
#>  1.529867  1.424100

## Test $$t$$ pour deux échantillons appariés

with(Copies,shapiro.test(Correcteur.A-Correcteur.B))
#>
#>  Shapiro-Wilk normality test
#>
#> data:  Correcteur.A - Correcteur.B
#> W = 0.97738, p-value = 0.7524
with(Copies,qqnorm(Correcteur.A-Correcteur.B))
with(Copies,qqline(Correcteur.A-Correcteur.B))

with(Copies,mean(Correcteur.A))
#> [1] 10.79333
with(Copies,mean(Correcteur.B))
#> [1] 11.37667
with(Copies,mean(Correcteur.A)-mean(Correcteur.B))
#> [1] -0.5833333
with(Copies,t.test(Correcteur.A-Correcteur.B))
#>
#>  One Sample t-test
#>
#> data:  Correcteur.A - Correcteur.B
#> t = -2.4078, df = 29, p-value = 0.02263
#> alternative hypothesis: true mean is not equal to 0
#> 95 percent confidence interval:
#>  -1.07881769 -0.08784898
#> sample estimates:
#>  mean of x
#> -0.5833333
with(Copies,t.test(Correcteur.A,Correcteur.B,paired=TRUE))
#>
#>  Paired t-test
#>
#> data:  Correcteur.A and Correcteur.B
#> t = -2.4078, df = 29, p-value = 0.02263
#> alternative hypothesis: true mean difference is not equal to 0
#> 95 percent confidence interval:
#>  -1.07881769 -0.08784898
#> sample estimates:
#> mean difference
#>      -0.5833333

## Analyse de la variance à un facteur

library(sageR)
data(Marque.Valeur)
str(Marque.Valeur)
#> 'data.frame':    90 obs. of  2 variables:
#>  $Marque: Factor w/ 3 levels "Marque 1","Marque 2",..: 1 1 1 1 1 1 1 1 1 1 ... #>$ Valeur: num  1.5 1.95 1.84 1.08 1.28 2.07 -0.33 1.9 1.94 3.69 ...
data(Marque.Valeur.large)
str(Marque.Valeur.large)
#> 'data.frame':    30 obs. of  3 variables:
#>  $Marque.1: num 1.5 1.95 1.84 1.08 1.28 2.07 -0.33 1.9 1.94 3.69 ... #>$ Marque.2: num  1.98 1.86 3.3 2.19 3.43 2.49 1.98 2.84 4 3.13 ...
#>  $Marque.3: num 4.46 6.49 4.69 3.23 4.53 3.61 3.45 2.91 4.25 4.12 ... round(with(Marque.Valeur,tapply(Valeur,Marque,mean)),3) #> Marque 1 Marque 2 Marque 3 #> 1.411 2.889 4.124 round(with(Marque.Valeur,tapply(Valeur,Marque,sd)),3) #> Marque 1 Marque 2 Marque 3 #> 0.870 1.051 0.939 round(with(Marque.Valeur,tapply(Valeur,Marque,var)),3) #> Marque 1 Marque 2 Marque 3 #> 0.756 1.105 0.882 library(ggplot2) ggplot(data=Marque.Valeur,aes(x=Marque,y=Valeur)) + geom_boxplot() + xlab("") options(contrasts = c("contr.sum","contr.sum")) lm.Marque.Valeur <- lm(Valeur~Marque,data=Marque.Valeur) anova(lm.Marque.Valeur) #> Analysis of Variance Table #> #> Response: Valeur #> Df Sum Sq Mean Sq F value Pr(>F) #> Marque 2 110.676 55.338 60.502 < 2.2e-16 *** #> Residuals 87 79.575 0.915 #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 -sum(coef(lm.Marque.Valeur)[2:3]) #> [1] 1.315667 res.model.MV <- residuals(lm.Marque.Valeur) shapiro.test(res.model.MV) #> #> Shapiro-Wilk normality test #> #> data: res.model.MV #> W = 0.98958, p-value = 0.6999 qqnorm(res.model.MV) qqline(res.model.MV) bartlett.test(res.model.MV,Marque.Valeur$Marque)
#>
#>  Bartlett test of homogeneity of variances
#>
#> data:  res.model.MV and Marque.Valeur$Marque #> Bartlett's K-squared = 1.0505, df = 2, p-value = 0.5914 if(!("ggiraphExtra" %in% installed.packages())){install.packages("ggiraphExtra")} library(ggiraphExtra) Marque.Valeur$Marque<-as.factor(rep(paste("M",1:3),rep(rep(30,3))))
oav.Marque.Valeur <- aov(Valeur~Marque,data=Marque.Valeur)
resHSD <- TukeyHSD(oav.Marque.Valeur)
resHSD
#>   Tukey multiple comparisons of means
#>     95% family-wise confidence level
#>
#> Fit: aov(formula = Valeur ~ Marque, data = Marque.Valeur)
#>
#> $Marque #> diff lwr upr p adj #> M 2-M 1 1.478333 0.8895222 2.067144 1.0e-07 #> M 3-M 1 2.712667 2.1238555 3.301478 0.0e+00 #> M 3-M 2 1.234333 0.6455222 1.823144 8.8e-06 ggHSD(resHSD) + ylab("Differences entre les niveaux moyens de Marque") + ggtitle("Intervalles de confiance globale 95%") #> Warning: guides(<scale> = FALSE) is deprecated. Please use guides(<scale> = #> "none") instead. C1=1+1/(3*(nlevels(Marque.Valeur$Marque)-1))*(sum(1/(table(Marque.Valeur$Marque)-1))-1/(nrow(Marque.Valeur)-nlevels(Marque.Valeur$Marque)))
C1
#> [1] 1.015326
1/C1*((nrow(Marque.Valeur)-nlevels(Marque.Valeur$Marque))*log(summary(oav.Marque.Valeur)[[1]]$Mean Sq[2])-sum((table(Marque.Valeur$Marque)-1)*log(with(Marque.Valeur,tapply(Valeur,Marque,var))))) #> [1] 1.050525 bartlett.test(res.model.MV,Marque.Valeur$Marque)
#>
#>  Bartlett test of homogeneity of variances
#>
#> data:  res.model.MV and Marque.Valeur\$Marque
#> Bartlett's K-squared = 1.0505, df = 2, p-value = 0.5914

## QQplots

oldpar <- par()

Chemin <- "~/Documents/Recherche/DeBoeck/Graphes/"
colmodel="cmyk"
library(VGAM)

par(mar = c(0.2, 0.2, 0.2, 0.2) + 0.1, mgp = c(2, 1, 0))
ech_norm <- rnorm(100)
qqnorm(ech_norm, ylab="",main="", xaxt="n", yaxt="n")
legend("bottomright",legend="(a)",cex=2,bty="n")

par(mar = c(0.2, 0.2, 0.2, 0.2) + 0.1, mgp = c(2, 1, 0))
ech_laplace <- rlaplace(100,1/sqrt(2))
qqnorm(ech_laplace, ylab="",main="", xaxt="n", yaxt="n")
legend("bottomright",legend="(b)",cex=2,bty="n")

par(mar = c(0.2, 0.2, 0.2, 0.2) + 0.1, mgp = c(2, 1, 0))
ech_unif <- runif(100,-2,2)
qqnorm(ech_unif, ylab="(c)",main="", xaxt="n", yaxt="n")
legend("bottomright",legend="(c)",cex=2,bty="n")

par(mar = c(0.2, 0.2, 0.2, 0.2) + 0.1, mgp = c(2, 1, 0))
ech_exp <- rexp(100,1)
qqnorm(ech_exp, ylab="",main="", xaxt="n", yaxt="n")
legend("bottomright",legend="(d)",cex=2,bty="n")

par(mar = c(0.2, 0.2, 0.2, 0.2) + 0.1, mgp = c(2, 1, 0))
ech_mel <- c(rnorm(50,0,1),rnorm(50,5,1))
qqnorm(ech_mel, ylab="", main="", xaxt="n", yaxt="n")
legend("bottomright",legend="(e)",cex=2,bty="n")

par(mar = c(0.2, 0.2, 0.2, 0.2) + 0.1, mgp = c(2, 1, 0))
ech_ext <- c(rnorm(99,0),5)
qqnorm(ech_ext, ylab="",main="", xaxt="n", yaxt="n")
legend("bottomright",legend="(f)",cex=2,bty="n")