Power Curves for t-test. Shoe Wear. α =.05, X 1,..., X n N(µ,1) > t.test(rnorm(20,mean=2,sd=2)) One-Sample t-test

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Dwight Noel Gray
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1 One-Samle -Tes Poer Crves for -es α =.05, X 1,..., X n N(µ,1) n=20.es(rnorm(20,mean=2,sd=2)) One Samle -es daa: rnorm(20, mean = 2, sd = 2) = 2.384, df = 19, -vale = alernaive hyohesis: re mean is no eqal o mean of x es(rnorm(20,mean=0,sd=2)) One Samle -es daa: rnorm(20, mean = 0, sd = 2) = , df = 19, -vale = alernaive hyohesis: re mean is no eqal o mean of x n=10 n= Shoe Wear library(mass) shoes $A [1] $B [1] layo(rbind(c(1,2),c(1,3))) boxlo(lis(a=shoes$a, B=shoes$B), noch=t, horizonal=t, boxex=.5) qqnorm(shoes$a); qqline(shoes$a) qqnorm(shoes$b); qqline(shoes$b) A B Paired -Tes.es(shoes$A, shoes$b, aired=t) Paired -es = , df = 9, -vale = mean of he differences es(shoes$A-shoes$B) One Samle -es daa: shoes$a - shoes$b = , df = 9, -vale = alernaive hyohesis: re mean is no eqal o mean of x

2 Paired vs. Unaired.es(shoes$A, shoes$b, aired=t) Paired -es = , df = 9, -vale = mean of he differences Non-normaliy of Traffic Daa library(mass) aach(traffic) qqnorm(y[limi=="yes"]); qqline(y[limi=="yes"]) qqnorm(y[limi=="no"]); qqline(y[limi=="no"]) shairo.es(y[limi=="yes"]) daa: y[limi == "yes"] W = , -vale = shairo.es(y[limi=="no"]) daa: y[limi == "no"] W = , -vale = es(shoes$A, shoes$b) # bad idea: sing aired=f Welch To Samle -es = , df = , -vale = Normaliy (?) of Log Traffic Daa qqnorm(log(y[limi=="yes"])) qqnorm(log(y[limi=="no"])) shairo.es(log(y[limi=="yes"])) daa: log(y[limi == "yes"]) W = , -vale = shairo.es(log(y[limi=="no"])) daa: log(y[limi == "no"]) W = , -vale = To-Samle -Tes ly.yes <- log(y[limi=="yes"]) ly.no <- log(y[limi=="no"]).es(ly.yes, ly.no) Welch To Samle -es daa: ly.yes and ly.no = , df = , -vale = var.es(ly.yes, ly.no) F es o comare o variances daa: ly.yes and ly.no F = , nm df = 68, denom df = 114, -vale = alernaive hyohesis: re raio of variances is no eqal o raio of variances es(ly.yes, ly.no, var.eqal=t) To Samle -es daa: ly.yes and ly.no = , df = 182, -vale =

3 Aroximae Z-Tess in R Insead of log-ransforming raffic daa, le s do an aroximae Z-es on he nransformed daa: Same saisic, so se same es fncion:.es(y[limi=="yes"],y[limi=="no"]) Welch To Samle -es daa: y[limi == "yes"] and y[limi == "no"] = , df = , -vale = alernaive hyohesis: re difference in means is no eqal o If yo an o be icky, se -vale based on Z insead of ( ): 2*( ,df= ,loer.ail=T) [1] *norm( ,loer.ail=T) [1] b his is all so aroximae anyay, hy no ake he more conservaive -based -vale? Recall ha exac -vale for log-ransformed daa assming normaliy and eqal variances as = Wilcoxon Signed-Rank Tes For one samle: x <- c(8.5,8.6,6.4,12.1,8.2,7.4,7.8,8.3,10.3,8.4) ilcox.es(x, =10, conf.in=t) Wilcoxon signed rank es daa: x V = 8, -vale = alernaive hyohesis: re is no eqal o (sedo)median 8.35 For aired daa: ilcox.es(shoes$a, shoes$b, aired=t) Wilcoxon signed rank es ih coniniy correcion V = 3, -vale = alernaive hyohesis: re is no eqal o 0 Warning message: Canno come exac -vale ih ies in: ilcox.es.defal(shoes$a, shoes$b, aired = T) ilcox.es(shoes$a-shoes$b) Wilcoxon signed rank es ih coniniy correcion daa: shoes$a - shoes$b V = 3, -vale = alernaive hyohesis: re is no eqal o 0 Warning message: Canno come exac -vale ih ies in: ilcox.es.defal(shoes$a - shoes$b) 39 Why I Love he Wilcoxon Tes Why I Love I Even More n=10 n=9 n=8 Normal Samle, sd=1, n=20 es (heoreical nder normaliy) es (silaed) Wilcoxon (silaed) n=7 n=6 n=5 Dis Samle, df=2, n= n=4 n=3 n=2 Mixre of 90% N(,sd=1) and 10% N(,sd=15), n=

4 To-Samle Wilcoxon Tes ilcox.es(y.yes,y.no) Wilcoxon rank sm es ih coniniy correcion daa: y.yes and y.no W = 2878, -vale = alernaive hyohesis: re is no eqal o 0 ilcox.es(log(y.yes),log(y.no)) Wilcoxon rank sm es ih coniniy correcion daa: log(y.yes) and log(y.no) W = 2878, -vale = alernaive hyohesis: re is no eqal o 0 Poer of Shairo Tes rejecs <- 0 for (i in 1:10000) { + x <- rnorm(30, mean=45, sd=20) + if (shairo.es(x)$.vale < 0.05) rejecs <- rejecs } rejecs [1] 530 rejecs <- 0 for (i in 1:10000) { + x <- rgamma(30, shae=2, rae=20) + if (shairo.es(x)$.vale < 0.05) rejecs <- rejecs } rejecs [1] 7513 rejecs <- 0 for (i in 1:10000) { + x < *r(30, df=4) + if (shairo.es(x)$.vale < 0.05) rejecs <- rejecs } rejecs [1] Poer of Shairo Tes Mligro Locaion Tess Normal Samle (Shairo =0.24) names(plangroh) [1] "eigh" "gro" levels(plangroh$gro) [1] "crl" "r1" "r2" oneay.es(eigh ~ gro, daa=plangroh) One-ay analysis of means (no assming eqal variances) daa: eigh and gro F = 5.181, nm df = 2.000, denom df = , -vale = Gamma Samle (Shairo = ) oneay.es(eigh ~ gro, daa=plangroh, var.eqal=t) Samle (Shairo =0.097) One-ay analysis of means daa: eigh and gro F = , nm df = 2, denom df = 27, -vale = smmary(aov(eigh ~ gro, daa=plangroh)) Df Sm Sq Mean Sq F vale Pr(F) gro * Residals Signif. codes: 0 *** ** 0.01 * krskal.es(eigh ~ gro, daa=plangroh) Krskal-Wallis rank sm es daa: eigh by gro Krskal-Wallis chi-sqared = , df = 2, -vale =

5 Tesing To-Way Coningencies able(r,sa404) sa404 r - n y chisq.es(able(r,sa404)) Pearson s Chi-sqared es daa: able(r, sa404) X-sqared = , df = 6, -vale = Warning message: Chi-sqared aroximaion may be incorrec in: chisq.es(able(r, sa404)) fisher.es(able(r,sa404)) Fisher s Exac Tes for Con Daa daa: able(r, sa404) -vale = alernaive hyohesis: o.sided Tesing To-Way Coningencies able(malab,sa404) sa404 malab - n y chisq.es(able(malab,sa404)) Pearson s Chi-sqared es daa: able(malab, sa404) X-sqared = , df = 6, -vale = Warning message: Chi-sqared aroximaion may be incorrec in: chisq.es(able(malab, sa404)) fisher.es(able(malab,sa404)) Fisher s Exac Tes for Con Daa daa: able(malab, sa404) -vale = alernaive hyohesis: o.sided Bigger Examle library(mass) names(melanoma) [1] "ime" "sas" "sex" "age" "year" [6] "hickness" "lcer" aach(melanoma) able(sex,lcer) lcer sex chisq.es(able(sex,lcer)) For Unaired (Indeenden) Indeenden, iid samles: A α = 0.05, Samles X 1,..., X 20 N(µ X,1) Y 1,..., Y 20 N(µ Y,1) Pearson s Chi-sqared es ih Yaes coniniy correcion daa: able(sex, lcer) X-sqared = , df = 1, -vale = fisher.es(able(sex,lcer)) Fisher s Exac Tes for Con Daa daa: able(sex, lcer) -vale = alernaive hyohesis: re odds raio is no eqal o odds raio Silaed Poer for Unaired Samles (n=20) difference in means aired (rong!) naired (okay) Welch (okay) 48 49

6 For Paired Daa (X 1, Y 1 ),...,(X 20, Y 20 ) iid bivariae normal. A α = 0.05, Posiive correlaion (sal) ρ = Silaed Poer for Paired Daa (n=20) difference in means Negaive correlaion (nsal) ρ = Silaed Poer for Paired Daa (n=20) aired (okay) naired (rong!) Welch (rong!) difference in means aired (okay) naired (rong!) Welch (rong!) 50 Locaion ess: Smmary of Tess.es: -ess for one samle, aired, o-samle (eqal variance and Welch s aroximaion). Can also be sed for aroximae Z-ess. ilcox.es: Wilcoxon signed-rank es for one samle or aired daa, Wilcoxon/Mann-Whiney rank-sm es for o samles. oneay.es: F-es for difference in means of o or more samles (eqal variance and Welch s aroximaion). krskal.es: Krskal-Wallis rank-sm es for difference in means of o or more samles. Caegorical daa ess: chisq.es: χ 2 -es for one-ay and ineracion in o-ay coningency ables. fisher.es: Fisher s exac es for ineracion in o-ay coningency ables. Oher ess: shairo.es: Shairo-Wilk es for H 0 : samle is normal. var.es: F-es for H 0 : samles have eqal variances. 51

Chapter 5. Two-Variable Regression: Interval Estimation and Hypothesis Testing

Chapter 5. Two-Variable Regression: Interval Estimation and Hypothesis Testing Chaper 5. Two-Variable Regression: Inerval Esimaion and Hypohesis Tesing Inerval Esimaion: Some Basic Ideas ( ) δ + δ where 0 < Pr < Lower Confidence Upper Confidence Confidence Level Significance Level