Linear models Analysis of Covariance

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1 Esben Budtz-Jørgensen April 22, 2008 Linear models Analysis of Covariance Confounding Interactions Parameterizations

2 Analysis of Covariance group comparisons can become biased if an important predictor of the response is distributed differently in the groups An unbiased analysis can be obtained in a multiple regression analysis with the group variable and the predictor as independent variables Examples: Comparison of blood pressure level in men and women when they are not equally fat Comparison of lung capacity in men and women when they are not of the same height 1

3 Lung Capacity, TLC 32 patients are planned to have a heart/lung transplantation TLC (Total Lung Capacity) determined by means of whole body plethysmography Is there a difference in lung capacity between men and women? OBS SEX AGE HEIGHT TLC 1 F F M F M M M

4 Box plots: total lung capacity female male height female male 3

5 TTEST PROCEDURE Variable: TLC Marginal comparisons SEX N Mean Std Dev Std Error F M Variances T DF Prob> T Unequal Equal For H0: Variances are equal, F = 1.22 DF = (15,15) Prob>F = Variable: HEIGHT SEX N Mean Std Dev Std Error F M Variances T DF Prob> T Unequal Equal For H0: Variances are equal, F = 1.30 DF = (15,15) Prob>F = Clear difference for both TLC and HEIGHT 4

6 Analysis of covariance Comparison of parallel regression lines MODEL: Y gi = α g + βx gi + ǫ gi g = 1, 2; i = 1,...,n g 5

7 What happens if we forget about x? MODEL: Y gi = α g + βx gi + ǫ gi g = 1, 2; i = 1,...,n g If x 1 x 2, the difference in group means (Ȳ2 Ȳ1) is biased. 6

8 Interaction The two lines can have different slopes. More general model: y gi = α g + β g x gi + ǫ gi g = 1, 2; i = 1,...,n g If β 1 β 2, the two covariates interact: Effect of height depends on sex Difference between males and females depends on height 7

9 Relationship between TLC and HEIGHT: 8

10 Relationship between log-transformed TLC and height, HEIGHT 9

11 Model specification: Model with interaction proc glm; class sex; model ltlc=sex height sex*height / solution; run; Or in SAS Analyst: ANOVA/Linear models choose ltlc as dependent choose height as a quantitative variable choose sex as a class variable under the Model button insert the cross -term 10

12 Output Dependent Variable: LTLC Sum of Mean Source DF Squares Square F Value Pr > F Model Error Corrected Total R-Square C.V. Root MSE LTLC Mean Source DF Type I SS Mean Square F Value Pr > F SEX HEIGHT HEIGHT*SEX Source DF Type III SS Mean Square F Value Pr > F SEX HEIGHT HEIGHT*SEX T for H0: Pr > T Std Error of Parameter Estimate Parameter=0 Estimate INTERCEPT B SEX F B M B... HEIGHT B HEIGHT*SEX F B M B... 11

13 Relationship between log-transformed TLC and height, HEIGHT 12

14 The interaction term was excluded Reduction of the model Dependent Variable: LTLC Sum of Mean Source DF Squares Square F Value Pr > F Model Error Corrected Total R-Square C.V. Root MSE LTLC Mean Source DF Type I SS Mean Square F Value Pr > F SEX HEIGHT Source DF Type III SS Mean Square F Value Pr > F SEX HEIGHT T for H0: Pr > T Std Error of Parameter Estimate Parameter=0 Estimate INTERCEPT B SEX F B M B... HEIGHT Note: Now the effect of sex has disappeared! 13

15 In this example we saw that Interpretation The observed difference in (log 10 ) lung function between females and males could be attributed to the difference in height A 95% confidence interval for log 10 -difference is ± = ( ,0.1125), corresponding to the interval (0.94, 1.30) for the ratio of lung capacity, i.e., men can have a 30% better lung function. It is also possible that Groups that appear to be equal in marginal analysis (e.g. blood pressure in men and women) show a difference after adjustment for important covariates (such as obesity) All variables with potential influence should be considered! 14

16 Example: Blood pressure vs. obesity and sex Marginal analysis indicates that there are no differences in blood pressure levels in males and females. However, when we adjust for the degree of obesity suddenly we can see a sex-difference. 15

17 with interaction: Model proc glm; class sex; model lbp=lobese sex sex*lobese / solution; run; 16

18 Output General Linear Models Procedure Dependent Variable: LBP Sum of Mean Source DF Squares Square F Value Pr > F Model Error Corrected Total Source DF Type I SS Mean Square F Value Pr > F LOBESE SEX LOBESE*SEX Source DF Type III SS Mean Square F Value Pr > F LOBESE SEX LOBESE*SEX T for H0: Pr > T Std Error of Parameter Estimate Parameter=0 Estimate INTERCEPT B SEX female B male B... LOBESE B LOBESE*SEX female B male B... 17

19 Re-parametrization proc glm; class sex; model lbp=sex sex*lobese / noint solution; run; General Linear Models Procedure Dependent Variable: LBP Sum of Mean Source DF Squares Square F Value Pr > F Model Error Uncorrected Total Source DF Type III SS Mean Square F Value Pr > F SEX LOBESE*SEX T for H0: Pr > T Std Error of Parameter Estimate Parameter=0 Estimate SEX female male LOBESE*SEX female male

20 The model is the same, 2 different parameterizations: 1. model lbp = lobese sex sex*lobese An intercept for the reference group (sex=1) An intercept difference from sex=0 to sex=1 An effect of lobese (slope) for the reference group A slope difference from sex=0 to sex=1 2. model lbp=sex sex*lobese / noint An intercept for each group (sex) A slope (lobese effect) for each group (sex) 19

21 Reduced model: no interaction (equal slopes) proc glm; class sex; model lbp=lobese sex / solution; run; 20

22 General Linear Models Procedure Reduced model, output Dependent Variable: LBP Sum of Mean Source DF Squares Square F Value Pr > F Model Error Corrected Total Source DF Type I SS Mean Square F Value Pr > F SEX LOBESE Source DF Type III SS Mean Square F Value Pr > F SEX LOBESE T for H0: Pr > T Std Error of Parameter Estimate Parameter=0 Estimate INTERCEPT B SEX female B male B... LOBESE NOTE: The X X matrix has been found to be singular and a generalized inverse was used to solve the normal equations. Estimates followed by the letter B are biased, and are not unique estimators of the parameters. 21

23 Conclusion The male level is higher than the female level (for fixed level of obesity), but remember this is on a log 10 -scale Confidence interval: ± = (0.0040, ) Back-transformed: (1.009, 1.126), i.e. the male level is between 1% og 12.6% above the female level 22