1 Experimental Statistics - week 2 Review: 2-sample t-tests paired t-tests Thursday: Meet in 15 Clements!! Bring Cody and Smith book

2 p- Value (observed value of t) p-value Suppose t = is observed from data for test above Note: “ Large negative values” of t make us believe alternative is true the probability of an observation as extreme or more extreme than the one observed when the null is true

3 Note: -- if p-value is less than or equal to  then we reject null at the  significance level -- the p-value is the smallest level of significance at which the null hypothesis would be rejected

4 Find the p-values for Examples 1 and 2

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6 Two Independent Samples Assumptions: Measurements from each population are –Mutually Independent  Independent within each sample  Independent between samples –Normally distributed (or the Central Limit Theorem can be invoked) Analysis differs based on whether the 2 populations have the same standard deviation

7 Two Cases Population standard deviations equal –Can obtain a better estimate of the common standard deviation by combining or “pooling” individual estimates Population standard deviations unequal –Must estimate each standard deviation –Very good approximate tests are available If Unsure, Do Not Assume Equal Standard Deviations

8 Equal Population Standard Deviations Test Statistic df = n 1 + n where

9 Behrens-Fisher Problem

10 Satterthwaite’s Approximate t Statistic (Approximate t df) (i.e. approximate t)

11 Often-Recommended Strategy for Tests on Means Test whether  1 =  2 (F-test ) –If the test is not rejected, use the 2-sample t statistics, assuming equal standard deviations –If the test is rejected, use Satterthwaite’s approximate t statistic NOTE: This is Not a good strategy –the F-test is highly susceptible to non-normality Recommended Strategy: –If uncertain about whether the standard deviations are equal, use Satterthwaite’s approximate t statistic

12 Example 3: Comparing the Mean Breaking Strengths of 2 Plastics Plastic A: Plastic B: Assumptions: Mutually independent measurements Normal distributions for measurements from each type of plastic Question: Question: Is there a difference between the 2 plastics in terms of mean breaking strength?

13 Example 3 - solution

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15 Design: 50 people: randomly assign 25 to go on diet and 25 to eat normally for next month. Assess results by comparing weights at end of 1 month. Diet: No Diet: Diet: No Diet: Run 2-sample t-test using guidelines we have discussed. Is this a good design? New diet – Is it effective?

16 Better Design: Randomly select subjects and measure them before and after 1-month on the diet. Subject Before After : : : n Difference 3 15 : -3 Procedure: Calculate differences, and analyze differences using a 1-sample test “Paired t-Test”

17 Example 4: International Gymnastics Judging Question: Do judges from a contestant’s country rate their own contestant higher than do foreign judges? Data:

18 Example 4 solution

19

Introduction to SAS Programming Language

21 Fertilizer Data Brand 1 Brand cm 54.0 cm A researcher studies the effect of two fertilizer brands on the growth of plants. Thirteen plants grown under identical conditions except that 7 plants are randomly selected to receive Brand 1 and the remaining 6 are fertilized using Brand 2. The data for this experiment are as follows where the outcome measurement is the height of the plant after 3 weeks of growth (you may assume the heights to be normally distributed):

22 The Fertilizer data set as SAS needs to see it A 51.0 A 53.3 A 55.6 A 51.0 A 55.5 A 53.0 A 52.1 B 54.0 B 56.1 B 52.1 B 56.4 B 54.0 B 52.9

23 Case 1: Data within SAS FILE : DATA one; INPUT brand$ height; DATALINES; A 51.0 A B 54.0 E 52.9 ; PROC TTEST; CLASS brand; VAR height; TITLE ‘Fertilizer Data – 2-sample t-test'; RUN; SAS file for FERTILIZER data

24 Brief Discussion of Components of the SAS File: DATA Step DATA STATEMENT - the first DATA statement names the data set whose variables are defined in the INPUT statement -- in the above, we create data set 'one' INPUT STATEMENT - 2 forms 1. Freefield - can be used when data values are separated by 1 or more blanks INPUT NAME $ AGE SEX $ SCORE; ($ indicates character variable) 2. Formatted - data occur in fixed columns INPUT NAME $ 1-20 AGE SEX $ 26 SCORE 28-30; DATALINES STATEMENT - used to indicate that the next records in the file contain the actual data and the semicolon after the data indicates the end of the data itself

25 SPECIFYING THE ANALYSIS SPECIFYING THE ANALYSIS -- PROC STATEMENTS GENERAL FORM PROC xxxxx; implies procedure is to be run on most recently created data set PROC xxxxx DATA = data set name; Note: I did not have to specify DATA=one in the above example Example PROCs: PROC REG - regression analysis PROC ANOVA - analysis of variance PROC GLM - general linear model PROC MEANS - basic statistics, t-test for H 0 :  PROC PLOT - plotting PROC TTEST - t-tests PROC UNIVARIATE - descriptive stats, box-plots, etc. PROC BOXPLOT - boxplots

26 PROC TTEST Proc TTEST data = fn ; Class … ; (specify the classification variable) Var … / options; ( specify the variable for which the means are compared) Run;

27 SAS Syntax MUSTEvery command MUST end with a semicolon –Commands can continue over two or more lines Variable names are 1-8 characters (letters and numerals, beginning with a letter or underscore), but no blanks or special characters –Note: values for character variables can exceed 8 characters Comments –Begin with *, end with ;

28 Titles and Labels TITLE ‘…’ ; –Up to 10 title lines: TITLE ‘include your title here’; –Can be placed in Data Steps or Procs LABEL name = ‘…’ ; –Can be in a DATA STEP or PROC PRINT –Include ALL labels, then a single ; Note: For class assignments, place descriptive titles and labels on the output.

29 Case 2: Data in External File : FILENAME f1 ‘complete directory/file specification’; FILENAME f1 ‘fertilizer.data'; DATA one; INFILE f1; INPUT brand$ height; PROC TTEST; CLASS brand; VAR height; TITLE ‘Fertilizer Data – 2-sample t-test'; RUN;

30 PC SAS on Campus Library BIC Student Center SAS Learning Edition $125