Logistic Regression Analyses in the Water Level Study

Transcription

1 Logistic Regression Analyses in the Water Level Study A. Introduction. 166 students participated in the Water level Study. 70 passed and 96 failed to correctly draw the water level in the glass. There were two main research questions: 1. Why was the passing rate so low? What factors affect passing? 2. There was a major difference in the proportion of females and males who passed? Can some of the variables in the study explain this? B. Frequency Tables: 1. Tabl e o f y by se x Frequency Percent Sex Row Pct Col Pct female male Total ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ % of females passed Fail (32/107) ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ % of males passed Pass (38/59) ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ Total Statistics for Table of y by sex Statistic DF Value Prob ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square <.0001 Likelihood Ratio Chi-Square < Table o f y by gravi ty Frequency Percent Gravity Score Row Pct Col Pct Total ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ Fail ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ Pass ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ Total Statistics for Table of y by gravity Page 2 Statistic DF Value Prob ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square <.0001 Likelihood Ratio Chi-Square <.0001 C. Logistic Regression of Pass/Fail in Water Level Study on Sex β 0 + β 1, for females : ln{π(sex)/[1-π(sex)]= β 0 + β 1 *(sex) =

2 β 0, for males options ls=72; Note 1: Females are coded 1 data sex; Males are coded 0 input sex r Note 2: Frequency counts are used cards; ; Proc Logistic ; r/n=sex ; output out=pred p=phat lower=lcl upper=ucl; proc print; run; Output: WORK.SEX Response Variable (Events) r Response Variable (Trials) n Number of Observations 2 Response Profile Ordered Binary Total Value Outcome Frequency 1 Event 70 2 Nonevent 96 and AIC SC Log L Page 3 Likelihood Ratio <.0001 Score <.0001 Wald <.0001 Test H 0 : No sex effect or H 0 : β 1 = 0 vs. H a : β 1 0. G 2 = = LRT Reject H 0 : No sex effect and conclude there is a statistically significant difference between females and males in proportion passing the task sex <.0001 Fitted : fitted logit(females) = = for females fitted logit(males) = for males

3 Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits sex Odds ratio (females vs. males) = s = Obs sex r n phat lcl ucl Odds ratio (males vs females): Pass Fail Males Females Odds Ratio = (38)(75)/(21)(32 = 4.24 = s D. Logistic Regression of Pass/Fail in Water Level Study on x = Gravity : ln π(x)/[1-π(x)]. SAS Program: Page 4 options ls=72; data gravity; input gravity r cards; ; Proc Logistic ; r/n=gravity ; output out=pred p=phat lower=lcl upper=ucl; proc print; run; WORK.GRAVITY Response Variable (Events) r Response Variable (Trials) n Number of Observations 6 Response Profile Ordered Binary Total Value Outcome Frequency 1 Event 70 2 Nonevent 96 and AIC SC Log L

4 Likelihood Ratio <.0001 Score <.0001 Wald <.0001 Test H 0 : No gravity effect or H 0 : β 1 = 0 vs. H a : β 1 0. G 2 = = LRT Reject H 0 : No gravity effect and conclude there is a statistically significant difference between gravity score and proportion passing the task. Page <.0001 gravity <.0001 Fitted : Estimated logit[π(x)] = x Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits gravity Odds o f passing the water l e ve l task incre ase by for e ach additional right answe r on gravity items. gravity r n r/n phat lcl ucl score (pass) (fail) A graph of observed and fitted proportions is Given above, right. How does the fit look? Y-Data Scatterplot of r/n, phat vs gravity Variable r/n phat E. Logistic regression of Pass/Fail on sex and gravity: gravity 4 5 options ls=72; Note: Females are coded 1 data water; Males are coded 2 input obs y sex gravity ; cards; ; Proc Logistic descending; Y=sex gravity; output out=pred p=phat lower=lcl upper=ucl; proc print;

5 Proc Logistic; Y=sex gravity; run; Note: descending not specified E1. Logistic Regression of Pass/Fail on Sex and Gravity : logit[π(sex, gravity)] = β 0 + β 1 *(sex) + β 2 *gravity Page 6 (β 0 + β 1 ) + β 2 *gravity, for females = (β 0 + 2β 1 )+ β 2 *gravity, for males WORK.WATER Response Variable y Number of Response Levels 2 Number of Observations 166 Probability modeled is y=1. and AIC SC Log L Likelihood Ratio <.0001 Score <.0001 Wald <.0001 Test H 0 : Sex and gravity together do not affect passing the water level task or H 0 : β 1 = β 2 = 0 vs. H a : at least one of the parameters is not 0. G 2 = = LRT Conclude the logistic regression of pass/fail on sex and gravity is statistically significant <.0001 sex gravity <.0001 Estimated logit(sex,gravity) = sex gravity. Note that sex is coded as 1 for females and 2 for males. Test the hypothesis that there is no gravity effect, adjusted for sex Page 7

6 Calculate the change in G 2 for the models with both variables included and with only sex. G 2 (sex, gravity) - G 2 (sex) = = 8.801, or calculate the change in - 2log likelihood: -2ln (sex) - [-2ln(sex, gravity) = = compare this value with the Wald chi-square Test the hypothesis that there is no sex effect, adjusted for gravity score: Calculate the change in G 2 for the models with both variables included and with only gravity. G 2 (sex, gravity) - G 2 (sex) = = , or calculate the change in - 2log likelihood: -2ln (gravity) - [-2ln(sex, gravity) = Compare this value with the Wald chi-square Point 95% Wald Effect Estimate Confidence Limits sex gravity Odds Ratio Estimates Predicted Values and Confidence Limits for Population Proportions: Obs obs y sex gravity _LEVEL_ phat lcl ucl Edited Fitted Values are given below; a plot of phat vs. gravity for females and for males is given in the graph. Row sex gravity phat lcl ucl phat Scatterplot of phat vs gravity gravity 5 sex 1 2 Page 8 E2. Logistic Regression of Pass/Fail on Sex, Gravity and Sex*Gravity (Interaction ) : logit[π(sex, gravity)] = β 0 + β 1 *(sex) + β 2 *gravity+β 3 *(sex*gravity) = (β 0 + β 1 ) + (β 2 + β 3 )gravity, for females

7 (β 0 + 2β 1 )+ (β 2 +2β 3 )gravity, for males WORK.PRED Response Variable y Number of Response Levels 2 Number of Observations 166 Ordered Total Value y Frequency Response Profile Probabil ity mode l ed is y= 0. and AIC SC Log L Likelihood Ratio <.0001 Score <.0001 Wald < sex gravity sex*gravity