Esben Budtz-Jørgensen April 22, 2008 Linear models Analysis of Covariance Confounding Interactions Parameterizations
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
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 35 149 3.40 2 F 11 138 3.41 3 M 12 148 3.80............... 29 F 20 162 8.05 30 M 25 180 8.10 31 M 22 173 8.70 32 M 25 171 9.45 2
Box plots: total lung capacity 4 6 8 female male height 140 160 180 female male 3
TTEST PROCEDURE Variable: TLC Marginal comparisons SEX N Mean Std Dev Std Error -------------------------------------------------------------------------- F 16 5.19812500 1.30082138 0.32520534 M 16 6.97687500 1.43801585 0.35950396 Variances T DF Prob> T Unequal -3.6693 29.7 0.0009 Equal -3.6693 30.0 0.0009 For H0: Variances are equal, F = 1.22 DF = (15,15) Prob>F = 0.7028 Variable: HEIGHT SEX N Mean Std Dev Std Error ----------------------------------------------------------------------- F 16 160.81250000 9.36816417 2.34204104 M 16 174.06250000 10.66126165 2.66531541 Variances T DF Prob> T Unequal -3.7344 29.5 0.0008 Equal -3.7344 30.0 0.0008 For H0: Variances are equal, F = 1.30 DF = (15,15) Prob>F = 0.6228 Clear difference for both TLC and HEIGHT 4
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
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
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
Relationship between TLC and HEIGHT: 8
Relationship between log-transformed TLC and height, HEIGHT 9
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
Output Dependent Variable: LTLC Sum of Mean Source DF Squares Square F Value Pr > F Model 3 0.27230446 0.09076815 13.05 0.0001 Error 28 0.19478293 0.00695653 Corrected Total 31 0.46708739 R-Square C.V. Root MSE LTLC Mean 0.582984 10.85524 0.08341 0.76835 Source DF Type I SS Mean Square F Value Pr > F SEX 1 0.13626303 0.13626303 19.59 0.0001 HEIGHT 1 0.13451291 0.13451291 19.34 0.0001 HEIGHT*SEX 1 0.00152852 0.00152852 0.22 0.6429 Source DF Type III SS Mean Square F Value Pr > F SEX 1 0.00210426 0.00210426 0.30 0.5867 HEIGHT 1 0.13597107 0.13597107 19.55 0.0001 HEIGHT*SEX 1 0.00152852 0.00152852 0.22 0.6429 T for H0: Pr > T Std Error of Parameter Estimate Parameter=0 Estimate INTERCEPT -.2190181620 B -0.62 0.5391 0.35221658 SEX F -.2810587157 B -0.55 0.5867 0.51102682 M 0.0000000000 B... HEIGHT 0.0060473650 B 2.99 0.0057 0.00201996 HEIGHT*SEX F 0.0014344422 B 0.47 0.6429 0.00306016 M 0.0000000000 B... 11
Relationship between log-transformed TLC and height, HEIGHT 12
The interaction term was excluded Reduction of the model Dependent Variable: LTLC Sum of Mean Source DF Squares Square F Value Pr > F Model 2 0.27077594 0.13538797 20.00 0.0001 Error 29 0.19631145 0.00676936 Corrected Total 31 0.46708739 R-Square C.V. Root MSE LTLC Mean 0.579712 10.70821 0.08228 0.76835 Source DF Type I SS Mean Square F Value Pr > F SEX 1 0.13626303 0.13626303 20.13 0.0001 HEIGHT 1 0.13451291 0.13451291 19.87 0.0001 Source DF Type III SS Mean Square F Value Pr > F SEX 1 0.00968023 0.00968023 1.43 0.2415 HEIGHT 1 0.13451291 0.13451291 19.87 0.0001 T for H0: Pr > T Std Error of Parameter Estimate Parameter=0 Estimate INTERCEPT -.3278068826 B -1.25 0.2198 0.26135206 SEX F -.0421012632 B -1.20 0.2415 0.03520676 M 0.0000000000 B... HEIGHT 0.0066723630 4.46 0.0001 0.00149683 Note: Now the effect of sex has disappeared! 13
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 0.0421 ± 2 0.0352 = ( 0.0283,0.1125), 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
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
with interaction: Model proc glm; class sex; model lbp=lobese sex sex*lobese / solution; run; 16
Output General Linear Models Procedure Dependent Variable: LBP Sum of Mean Source DF Squares Square F Value Pr > F Model 3 0.05583810 0.01861270 6.30 0.0006 Error 98 0.28952497 0.00295434 Corrected Total 101 0.34536306 Source DF Type I SS Mean Square F Value Pr > F LOBESE 1 0.03809379 0.03809379 12.89 0.0005 SEX 1 0.01597238 0.01597238 5.41 0.0221 LOBESE*SEX 1 0.00177193 0.00177193 0.60 0.4405 Source DF Type III SS Mean Square F Value Pr > F LOBESE 1 0.03920980 0.03920980 13.27 0.0004 SEX 1 0.01252714 0.01252714 4.24 0.0421 LOBESE*SEX 1 0.00177193 0.00177193 0.60 0.4405 T for H0: Pr > T Std Error of Parameter Estimate Parameter=0 Estimate INTERCEPT 2.087171366 B 165.93 0.0001 0.01257865 SEX female -0.039290663 B -2.06 0.0421 0.01908066 male 0.000000000 B... LOBESE 0.227981122 B 1.73 0.0863 0.13158758 LOBESE*SEX female 0.123097524 B 0.77 0.4405 0.15894836 male 0.000000000 B... 17
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 4 449.803216 112.450804 38062.97 0.0001 Error 98 0.289525 0.002954 Uncorrected Total 102 450.092741... Source DF Type III SS Mean Square F Value Pr > F SEX 2 141.530202 70.765101 23952.96 0.0001 LOBESE*SEX 2 0.054676 0.027338 9.25 0.0002 T for H0: Pr > T Std Error of Parameter Estimate Parameter=0 Estimate SEX female 2.047880703 142.73 0.0001 0.01434744 male 2.087171366 165.93 0.0001 0.01257865 LOBESE*SEX female 0.351078645 3.94 0.0002 0.08915879 male 0.227981122 1.73 0.0863 0.13158758 18
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
Reduced model: no interaction (equal slopes) proc glm; class sex; model lbp=lobese sex / solution; run; 20
General Linear Models Procedure Reduced model, output Dependent Variable: LBP Sum of Mean Source DF Squares Square F Value Pr > F Model 2 0.05406617 0.02703308 9.19 0.0002 Error 99 0.29129690 0.00294239 Corrected Total 101 0.34536306... Source DF Type I SS Mean Square F Value Pr > F SEX 1 0.00116215 0.00116215 0.39 0.5311 LOBESE 1 0.05290402 0.05290402 17.98 0.0001 Source DF Type III SS Mean Square F Value Pr > F SEX 1 0.01597238 0.01597238 5.43 0.0218 LOBESE 1 0.05290402 0.05290402 17.98 0.0001 T for H0: Pr > T Std Error of Parameter Estimate Parameter=0 Estimate INTERCEPT 2.081052655 B 213.05 0.0001 0.00976800 SEX female -0.027765105 B -2.33 0.0218 0.01191694 male 0.000000000 B... LOBESE 0.312347032 4.24 0.0001 0.07366198 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
Conclusion The male level is 0.0278 higher than the female level (for fixed level of obesity), but remember this is on a log 10 -scale Confidence interval: 0.0278 ± 2 0.0119 = (0.0040, 0.0516) Back-transformed: (1.009, 1.126), i.e. the male level is between 1% og 12.6% above the female level 22