Happiness comes not from material wealth but less desire.

Happiness comes not from material wealth but less desire.

Statistical Package Usage Topic: Tests for two samples By Prof Kelly Fan, Cal State Univ, East Bay

Example: Mortar Strength Unmodified modified16.85 17.5016.40 17.6317.21 18.2516.35 18.0016.52 17.8617.04 17.7516.96 18.2217.15 17.9016.59 17.9616.57 18.15

SAS Output The TTEST Procedure Statistics Lower CL Upper CL Lower CL Upper CL Variable FORMULATION N Mean Mean Mean Std Dev Std Dev Std Dev Std Err STRENGTH M 10 16.538 16.764 16.99 0.2177 0.3164 0.5777 0.1001 STRENGTH U 10 17.745 17.922 18.099 0.1705 0.2479 0.4526 0.0784 STRENGTH Diff (1-2) -1.425 -1.158 -0.891 0.2148 0.2843 0.4204 0.1271 Ho: The mean strength of the two formulations are the same H1: They are different

Types of Errors H0 true H0 false Type II Error, or “ Error” Good! (Correct!) we accept H0 Type I Error, or “ Error” Good! (Correct) we reject H0

= Probability of Type I error = P(rej. H0|H0 true) •  = Probability of Type II error= P(acc. H0|H0 false) We often preset . The value of  depends on the specifics of the H1 (and most often in the real world, we don’t know these specifics).

EXAMPLE: H0 : < 100 H1 :  >100 Suppose the Critical Value = 141: X  =100 C=141

 = P (X < 141/= 150) = .3594  = 150 What is ? 141  = 150  = 160 These are values corresp.to a value of 25 for the Std. Dev. of X  = P (X < 141/= 160) = .2236  = 160 141  = 170  = P (X < 141/= 170) = .1230  = 170 141  = 180  = P (X < 141/= 180) = .0594  = P (X < 141|H0 false)  = 180 141

Note: Had  been preset at .025 (instead of .05), C would have been 149 (and  would be larger); had  been preset at .10, C would have been 132 and  would be smaller. and“trade off”.

P Value • The probability of seeing as extreme as or more extreme than what we observe, assuming Ho is true. • The smaller the p-value is, the stronger the evidence against Ho is.

One Population Non-parametric test

Two independent samples Non-parametric test

Normality Tests Tests for Normality Test --Statistic--- -----p Value------ Shapiro-Wilk W 0.918255 Pr < W 0.0917 Kolmogorov-Smirnov D 0.134926 Pr > D >0.1500 Cramer-von Mises W-Sq 0.081542 Pr > W-Sq 0.1936 Anderson-Darling A-Sq 0.537514 Pr > A-Sq 0.1503 SAS: PROC UNIVARIATE DATA=** NORMAL; Note: For demonstration purpose, here is the result for testing normality when treating 2 samples as a whole. But you should do the normality test for both samples separately.

Equal-variance Tests SAS: PROC TTEST DATA=** ;

Non-parametric Tests • One sample: Wilcoxon Signed-Rank Test (Included in the output of Proc Univariate) • Two samples: Wilcoxon Rank Sum Test (See p. 191) • Three or more samples: Kruskal-Wallis Test (Later)

Paired Samples • If the same set of sources are used to obtain data representing two populations, the two samples are called paired. The data might be paired: • As a result of the data from certain “before” and “after” studies • From matching two subjects to form “matched pairs”

Example: Repeated Measures • Each subject is measured twice: • Before treatment (control value) • After treatment (treatment value)

Tests for Paired Samples • Calculate the pair differences • Proceed as in one sample case Notes: • SAS: all variables must be included in data • SPSS: create/calculate all variables we need Notes: • SAS and SPSS can also conduct paired t-tests directly

Happiness comes not from material wealth but less desire.