1. Linear statistical models 2009 Models for continuous, binary and binomial responses  Simple linear models regarded as special cases of GLMs  Simple linear regression  One-way ANOVA  Two-way ANOVA with or without interaction effects  Some useful continuous distributions  Binary and binomial responses

2. Linear statistical models 2009 A simple linear regression model

3. Linear statistical models 2009 GENMOD implementation of simple linear regression proc genmod data=linear.heartrate; model heart_rate = temp /dist=normal link=identity; run; Analysis of Parameter Estimates Standard Wald 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept 1 2.1389 1.6906 -1.1746 5.4524 1.60 0.2058 Temp 1 1.7750 0.1502 1.4806 2.0694 139.63 <.0001 Scale 1 2.3271 0.5485 1.4662 3.6936 NOTE: The scale parameter was estimated by maximum likelihood.

4. Linear statistical models 2009 Comparison of GENMOD and MINITAB’s simple linear regression GENMOD Analysis of Parameter Estimates Standard Wald 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept 1 2.1389 1.6906 -1.1746 5.4524 1.60 0.2058 Temp 1 1.7750 0.1502 1.4806 2.0694 139.63 <.0001 Scale 1 2.3271 0.5485 1.4662 3.6936 NOTE: The scale parameter was estimated by maximum likelihood. MINITAB Regression Analysis: Heart_rate versus Temp The regression equation is Heart_rate = 2.14 + 1.77 Temp Predictor Coef SE Coef T P Constant 2.139 1.917 1.12 0.301 Temp 1.7750 0.1703 10.42 0.000 S = 2.63869 R-Sq = 93.9% R-Sq(adj) = 93.1%

5. Linear statistical models 2009 Comparison of GENMOD and MINITAB’s simple linear regression  The point estimates of the fitted line are identical  The deviance in GENMOD is equal to the error sum of squares  The estimates of the standard deviation are different  The Wald-tests and the t-tests are different

6. Linear statistical models 2009 One-way ANOVA Measurement of hardness for nine groups of samples (3 levels of Zr, 3 temperature levels)

7. Linear statistical models 2009 GENMOD implementation of one-way ANOVA proc genmod data=linear.hardness; class zr_content temperature sample; model hardness_Gpa = sample /dist=normal link=identity; run; Analysis Of Parameter Estimates Standard Wald 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept 1 35.9016 0.4357 35.0476 36.7555 6789.78 <.0001 Sample 1 1 -6.8011 0.6130 -8.0024 -5.5997 123.11 <.0001 Sample 2 1 -6.8303 0.6195 -8.0446 -5.6161 121.56 <.0001 Sample 3 1 -8.1457 0.6130 -9.3471 -6.9443 176.61 <.0001 Sample 4 1 -13.4144 0.6195 -14.6286 -12.2002 468.86 <.0001 Sample 5 1 -8.6257 0.4800 -9.5665 -7.6850 322.95 <.0001 Sample 6 1 -10.4443 0.6099 -11.6396 -9.2490 293.30 <.0001 Sample 7 1 -8.5459 0.6162 -9.7535 -7.3382 192.36 <.0001 Sample 8 1 -3.1868 0.6565 -4.4735 -1.9001 23.56 <.0001 Sample 9 0 0.0000 0.0000 0.0000 0.0000.. Scale 1 2.9870 0.0871 2.8211 3.1627 NOTE: The scale parameter was estimated by maximum likelihood.

8. Linear statistical models 2009 GENMOD implementation of one-way ANOVA Standard Wald 95% Confidence Chi- GENMOD Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept 1 35.9016 0.4357 35.0476 36.7555 6789.78 <.0001 Sample 1 1 -6.8011 0.6130 -8.0024 -5.5997 123.11 <.0001 Sample 2 1 -6.8303 0.6195 -8.0446 -5.6161 121.56 <.0001 Sample 3 1 -8.1457 0.6130 -9.3471 -6.9443 176.61 <.0001 Sample 4 1 -13.4144 0.6195 -14.6286 -12.2002 468.86 <.0001 Sample 5 1 -8.6257 0.4800 -9.5665 -7.6850 322.95 <.0001 Sample 6 1 -10.4443 0.6099 -11.6396 -9.2490 293.30 <.0001 Sample 7 1 -8.5459 0.6162 -9.7535 -7.3382 192.36 <.0001 Sample 8 1 -3.1868 0.6565 -4.4735 -1.9001 23.56 <.0001 Sample 9 0 0.0000 0.0000 0.0000 0.0000.. Scale 1 2.9870 0.0871 2.8211 3.1627 MINTAB ANOVA Pooled StDev Level N Mean StDev ------+---------+---------+---------+--- 1 48 29.100 2.770 (-*-) 2 46 29.071 2.605 (--*-) 3 48 27.756 1.777 (-*--) 4 46 22.487 2.842 (-*-) 5 220 27.276 2.699 (*) 6 49 25.457 3.385 (-*-) 7 47 27.356 4.465 (-*--) 8 37 32.715 3.815 (--*-) 9 47 35.902 3.236 (-*-) ------+---------+---------+---------+--- 24.0 28.0 32.0 36.0 Pooled StDev = 3.010

9. Linear statistical models 2009 GENMOD implementation of two-way ANOVA proc genmod data=linear.hardness; class zr_content temperature sample; model hardness_Gpa = zr_content temperature zr_content * temperature/dist=normal link=identity; run; Analysis Of Parameter Estimates Standard Wald 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept 1 27.7559 0.4311 26.9108 28.6009 4144.59 <.0001 Zr_content 0.17 1 8.1457 0.6130 6.9443 9.3471 176.61 <.0001 Zr_content 0.5 1 -2.2986 0.6066 -3.4875 -1.1097 14.36 0.0002 Zr_content 1 0 0.0000 0.0000 0.0000 0.0000.. Temperature 400 1 1.3446 0.6097 0.1496 2.5397 4.86 0.0274 Temperature 800 1 1.3154 0.6163 0.1074 2.5233 4.56 0.0328 Temperature 1000 0 0.0000 0.0000 0.0000 0.0000.. Zr_conten*Temperatur 0.17 400 1 -9.8905 0.8668 -11.5895 -8.1915 130.18 <.0001 Zr_conten*Temperatur 0.17 800 1 -4.5022 0.9004 -6.2670 -2.7373 25.00 <.0001 Zr_conten*Temperatur 0.17 1000 0 0.0000 0.0000 0.0000 0.0000.. Zr_conten*Temperatur 0.5 400 1 -4.3147 0.8648 -6.0096 -2.6199 24.90 <.0001 Zr_conten*Temperatur 0.5 800 1 0.5032 0.7762 -1.0181 2.0245 0.42 0.5168 Zr_conten*Temperatur 0.5 1000 0 0.0000 0.0000 0.0000 0.0000.. Zr_conten*Temperatur 1 400 0 0.0000 0.0000 0.0000 0.0000.. Zr_conten*Temperatur 1 800 0 0.0000 0.0000 0.0000 0.0000.. Zr_conten*Temperatur 1 1000 0 0.0000 0.0000 0.0000 0.0000.. Scale 1 2.9870 0.0871 2.8211 3.1627

10. Linear statistical models 2009 The gamma distribution Expected value:  Variance:  2

11. Linear statistical models 2009 The  2 distribution Expected value: p Variance: 2p Special case of gamma distribution Sum of independent squared standard normal distributions

12. Linear statistical models 2009 A model of the mean of a gamma distribution proc genmod data=linear.clottingtime; model clotting_time = lconc agent lconc * agent/dist=gamma link=power(-1) residuals; output out=linear.clottingout resdev=resdev pred=pred; run;

13. Linear statistical models 2009 Binary and binomial responses The response probabilities are modelled as functions of the predictors Link functions: the probit link: the logit link: the log-log link: