ࡱ> { ^bjbjBrBr 4  V%<<<<<PPP8P!%"$$$$$$$$&)$<$<<$P#P#P#<<$P#$P#P#P#!DP#$$0!%P#/*!/*P#/*<P#<>,P#$A$$n"!%/* : ANCOVA Examples Using SAS This handout illustrates how to fit an ANCOVA model using a regression model with dummy variables and an interaction term in SAS. We also illustrate the same model fit using Proc GLM. We examine a dataset that illustrates the relationship between Height and Weight in a group of 237 teen-aged boys and girls. Import the Data from SPSS and check the values We first import the HTWT.SAV dataset from SPSS, get simple descriptive statistics for the numeric variables, and look at oneway frequencies for categorical variables. there are 237 participants who are from 13.9 to 25 years old. It is a cross-sectional study, with each participant having one observation. We can use this data set to examine the relationship of participants height to their age and sex. PROC IMPORT OUT= WORK.htwt DATAFILE= "C:\Documents and Settings\kwelch\Desktop\b510\htwt.sav" DBMS=SAV REPLACE; RUN; title "Contents of HTWT Data Set"; proc contents data=htwt; run; Contents of HTWT Data Set The CONTENTS Procedure Data Set Name WORK.HTWT Observations 237 Member Type DATA Variables 4 Engine V9 Indexes 0 Created Thursday, April 16, Observation Length 32 2009 08:08:45 AM Last Modified Thursday, April 16, Deleted Observations 0 2009 08:08:45 AM Protection Compressed NO Data Set Type Sorted NO Label Data Representation WINDOWS_32 Encoding wlatin1 Western (Windows) Alphabetic List of Variables and Attributes # Variable Type Len Format Informat Label 2 AGE Num 8 F5.2 AGE 3 HEIGHT Num 8 F5.2 HEIGHT 1 SEX Char 8 $8. $8. SEX 4 WEIGHT Num 8 F6.2 WEIGHT title "Descriptive Statistics for HTWT Data Set"; proc means data=htwt; run; Descriptive Statistics for HTWT Data Set The MEANS Procedure Variable Label N Mean Std Dev Minimum Maximum ----------------------------------------------------------------------------- AGE AGE 237 16.4430380 1.8425767 13.9000000 25.0000000 HEIGHT HEIGHT 237 61.3645570 3.9454019 50.5000000 72.0000000 WEIGHT WEIGHT 237 101.3080169 19.4406980 50.5000000 171.5000000 ---------------------------------------------------------------------------- title "Oneway Frequency Tabulation for Sex for HTWT Data Set"; proc freq data=htwt; tables sex; run; Oneway Frequency Tabulation for Sex for HTWT Data Set The FREQ Procedure Cumulative Cumulative SEX Frequency Percent Frequency Percent ------------------------------------------------------------- f 111 46.84 111 46.84 m 126 53.16 237 100.00 ANCOVA Models Using Proc Reg When we use Proc Reg to fit an ANCOVA model involving interactions, and dummy variables, we must first create these variables in a data step. In the data step below, we create a dummy variable for FEMALE, and another variable, FEM_AGE, which represents the interaction between FEMALE and AGE. We will be using these new variables in fitting our ANCOVA models. /*Create a new data set with new variables*/ data htwt2; set htwt; /*Create dummy variables for female*/ if sex="f" then female=1; if sex="m" then female=0; /*Create interaction*/ fem_age = female * age; run; Now we generate a scatter plot with HEIGHT as the Y and AGE as the X , with a separate regression line for males and females. title "Regression Plot of Height by Age"; title2 "For Males and Females"; proc sgplot data=htwt2; where age <= 19; reg y=height x=age / group=sex; run;  Older participants are taller, for both males and females, but it appears that Males have a slightly lower height at the youngest age, and a steeper slope than females. By the age of 19, males are about five inches taller than females, on average. ANCOVA Model with Original Variables: Next, we fit an ANCOVA model, representing the relationships shown on the graph. We include the main effects of sex (FEMALE), AGE, and their interaction. Note that we include the interaction created from the original AGE variable in this model. We again use a where statement to restrict the analysis to those who are less than or equal to 19 years old. We use the clb option to get a 95% confidence interval for each of the parameters in the model. The model that we are fitting is:  EMBED Equation.DSMT4  title "ANCOVA for Males and Females"; title2 "Relationship of Height to Age"; proc reg data=htwt2; where age <=19; model height = female age fem_age / clb; run; quit; We interpret the overall significance by looking at the Analysis of Variance table. We see that the model has 3 degrees of freedom, corresponding to the 3 predictors included in the model. The overall model is significant, F(3,215) = 60.93, p<0.001. The adjusted R-square is 0.4520. Model: MODEL1 Dependent Variable: HEIGHT Number of Observations Read 219 Number of Observations Used 219 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 3 1432.63813 477.54604 60.93 <.0001 Error 215 1684.95730 7.83701 Corrected Total 218 3117.59543 Root MSE 2.79947 R-Square 0.4595 Dependent Mean 61.00457 Adj R-Sq 0.4520 Coeff Var 4.58895 We examine the parameter estimates in the output below. Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| 95% Confidence Limits Intercept 1 28.88281 2.87343 10.05 <.0001 23.21911 34.54650 female 1 13.61231 4.01916 3.39 0.0008 5.69031 21.53432 AGE 1 2.03130 0.17764 11.44 <.0001 1.68117 2.38144 fem_age 1 -0.92943 0.24782 -3.75 0.0002 -1.41791 -0.44096 We first look at the parameter estimate for the interaction term. The interaction term, (3 (estimated to be -.92942) represents the difference in the slope of the regression line for females vs. the reference category (males). The estimated slope for AGE for females is -.92943 less than for males. and this difference in slopes is significant (t215 = -3.75, p = .0002). We next look at the parameter estimate for the main effect of AGE. The parameter for AGE , (2 (estimated to be 2.03) represents the slope for the reference group, males. This means that the average height of males increases by about 2 inches for each year of age. The estimated slope for males is significant (t215 = 11.44, p < 0.0001). The parameter estimate for FEMALE, (1(estimated to be 13.61) represents the difference in the intercept for females vs. males, i.e., the difference in predicted average height for females vs. males when AGE = 0. This is clearly a non-interpretable value, because AGE = 0 is outside the range of our data. The INTERCEPT, (0, represents the intercept for males (the reference group). The parameter estimate is 28.88, which means we estimate that males would have a height of about 29 inches at age 0, if we extrapolate our regression line to zero. Again, we have no interest in interpreting this intercept. This whole exercise is a cautionary tale in parameter interpretation. When you have an interaction in a model, it is tricky to interpret the main effects in the model. The main effects have to be interpreted with a great deal of care, if at all, in a model when there is an interaction in the model. ANCOVA Model with Centered Age: One way to help in the interpretation of the coefficients in a model like this is to center the continuous covariate, AGE, and then create an interaction term between centered age (CENTAGE) and FEMALE. Our new interaction variable will be called FEM_CENTAGE. In the SAS code below, we center age at 16.5 years, which is the approximate mean of AGE. This is like shifting the X-Axis in our model, so that the value of 0 for CENTAGE represents 16.5 years of actual age. data htwt2; set htwt2; /*Center age at 16.5 years*/ centage = age - 16.5; /*Create interactions*/ fem_centage = female * centage; run; We now refit the ANCOVA model using centered age, CENTAGE, and the interaction between female and centered age, FEM_CENTAGE, as predictors, rather than our original variables. title "ANCOVA for Males and Females"; title2 "Relationship of Height to Centered Age"; proc reg data=htwt2; where age <=19; model height = female centage fem_centage / clb; plot rstudent. * predicted.; output out=outreg p=predict r=resid rstudent=rstud; run; Note that the Analysis of Variance table and the model R-Square in the output below are the same as for the previous model. However, the parameter estimates are different. ANCOVA for Males and Females Relationship of Height to Centered Age The REG Procedure Model: MODEL1 Dependent Variable: HEIGHT Number of Observations Read 219 Number of Observations Used 219 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 3 1432.63813 477.54604 60.93 <.0001 Error 215 1684.95730 7.83701 Corrected Total 218 3117.59543 Root MSE 2.79947 R-Square 0.4595 Dependent Mean 61.00457 Adj R-Sq 0.4520 Coeff Var 4.58895 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| 95% Confidence Limits Intercept 1 62.39929 0.26902 231.95 <.0001 61.86904 62.92955 female 1 -1.72336 0.38917 -4.43 <.0001 -2.49044 -0.95629 centage 1 2.03130 0.17764 11.44 <.0001 1.68117 2.38144 fem_centage 1 -0.92943 0.24782 -3.75 0.0002 -1.41791 We now look at the parameter estimates in detail. We can see that the estimated interaction term, which represents the difference in slope for females vs. males (estimated to be -.92943), is the same as in the previous model. The coefficient for CENTAGE, which represents the slope for the reference category (males), is the same as the coefficient for AGE in the previous model, and is 2.03 inches per year. However, the estimated values for the variables FEMALE and the INTERCEPT are different than in the previous model. We see that the estimated effect of FEMALE is now -1.72. We can interpret this as the estimated difference in the average height of females vs. males when they are 16.5 years old (i.e. when CENTAGE is zero). In other words, females are about 1.7 inches shorter than males, on average, at age 16.5 years, and this difference in height is significant (t215 = -4.43, p < 0.001). This is a meaningful difference, which has a nice interpretation. The INTERCEPT in this model tells us the estimated average height of males when CENTAGE is zero (meaning that their actual age is 16.5 years). The estimated mean height for males at age 16.5 is 62.4 inches. This is also a meaningful value that we can interpret with confidence, because it is in the middle of the range of our data for AGE. Moral of the story: It is often helpful to center continuous variables in a regression model. It helps in interpreting the intercept in the model, and can also help in interpreting the main effects of variables that are included in interactions. When one centers the continuous variable, the interaction term is computed by multiplying the dummy variable for FEMALE times the Centered version of the continuous variable. ANCOVA Model Using Proc GLM We now refit the model using CENTAGE and SEX as predictors, but using Proc GLM. The advantage of using this procedure is that we dont need to create dummy variables for our categorical predictors, and the interaction terms do not need to be created in advance. The categorical variable(s ) (e.g., SEX) are listed in the Class Statement in SAS. The solution option is used to request that SAS print out the parameter estimates from the model. This option is not necessary, but is used for comparison with the parameter estimates from Proc Reg. title "ANCOVA model using GLM"; title "Relationship of Height to Centered AGE"; proc glm data=htwt2; where age <=19; class sex; model height = sex centage sex*centage / solution; run; quit; Relationship of Height to Centered AGE The GLM Procedure Class Level Information Class Levels Values SEX 2 f m Number of Observations Read 219 Number of Observations Used 219 Dependent Variable: HEIGHT Sum of Source DF Squares Mean Square F Value Pr > F Model 3 1432.638133 477.546044 60.93 <.0001 Error 215 1684.957300 7.837011 Corrected Total 218 3117.595434 R-Square Coeff Var Root MSE HEIGHT Mean 0.459533 4.588945 2.799466 61.00457 In the output below, the Type I SS shows the effect of each predictor in the model, sequentially. That is, the effect of SEX is evaluated without controlling for the other predictors. The effect of AGE is evaluated with only SEX in the model, and the effect of the CENTAGE by SEX interaction is evaluated, after adjusting for both main effects. The total of the Type I SS is equal to the total model SS. Source DF Type I SS Mean Square F Value Pr > F SEX 1 89.225774 89.225774 11.39 0.0009 centage 1 1233.182326 1233.182326 157.35 <.0001 centage*SEX 1 110.230033 110.230033 14.07 0.0002 The Type III SS below shows the effect of each predictor in the model, controlling for all other effects. The Type III SS is sometimes called the regression sum of squares or partial sum of squares. In this case, the total of the Type III SS does not equal the total model SS. Source DF Type III SS Mean Square F Value Pr > F SEX 1 153.684358 153.684358 19.61 <.0001 centage 1 1252.650148 1252.650148 159.84 <.0001 centage*SEX 1 110.230033 110.230033 14.07 0.0002 Notice that we get the same parameter estimates using Proc GLM as we did in Proc Reg. By default, Proc GLM overparameterizes the model, including a parameter for each level of SEX. The parameter estimate for the highest level of SEX is set to zero, which has the effect in this case of making males the reference category, as we had when we fit the model using Proc Reg. Switching the reference categories would not change the overall model fit, but would change the parameter estimates. Although the parameters are not uniquely estimable in this overparameterized model, we can interpret the parameter estimates, knowing the convention that SAS uses for the parameters in the model. Standard Parameter Estimate Error t Value Pr > |t| Intercept 62.39929305 B 0.26902176 231.95 <.0001 SEX f -1.72336221 B 0.38916785 -4.43 <.0001 SEX m 0.00000000 B . . . centage 2.03130229 B 0.17763755 11.44 <.0001 centage*SEX f -0.92943492 B 0.24782450 -3.75 0.0002 centage*SEX m 0.00000000 B . . . NOTE: The X'X matrix has been found to be singular, and a generalized inverse was used to solve the normal equations. Terms whose estimates are followed by the letter 'B' are not uniquely estimable. We now examine the plot of studentized residuals vs. predicted values, which we requested as part of the Proc Reg commands, to check for equality of variances,:  We can use Proc Univariate to check the distribution of the residuals for normality, and we see that the residuals appear to be quite normal.   Separate Regression Models for Males and Females We now fit separate regression models for females and males, using CENTAGE as the predictor of HEIGHT for each sex. To do this, we first sort by SEX, and then fit the regression models by SEX, using a by statement. And, remembering to select only cases with AGE<=19! We can see from the output below that the parameter estimates for females and males match what we had derived from the ANCOVA models, with Centered Age as a predictor. The advantage of the ANCOVA model is that we get a direct test of whether the slope for AGE is the same for females and males, whereas in the individual regression models, we do not. proc sort data = htwt2; by sex; run; title "Separate Regressions for Females and Males"; proc reg data = htwt2; where age <=19; by sex; model height = centage; run; quit; Separate Regressions for Females and Males ----------------------------------- SEX=f ----------------------------------- The REG Procedure Model: MODEL1 Dependent Variable: HEIGHT HEIGHT Number of Observations Read 103 Number of Observations Used 103 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 1 318.63384 318.63384 38.46 <.0001 Error 101 836.85005 8.28564 Corrected Total 102 1155.48388 Root MSE 2.87848 R-Square 0.2758 Dependent Mean 60.32718 Adj R-Sq 0.2686 Coeff Var 4.77145 Parameter Estimates Parameter Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 60.67593 0.28915 209.84 <.0001 centage 1 1.10187 0.17768 6.20 <.0001 Separate Regressions for Females and Males ----------------------------------- SEX=m ----------------------------------- The REG Procedure Model: MODEL1 Dependent Variable: HEIGHT HEIGHT Number of Observations Read 116 Number of Observations Used 116 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 1 1024.77852 1024.77852 137.75 <.0001 Error 114 848.10725 7.43954 Corrected Total 115 1872.88578 Root MSE 2.72755 R-Square 0.5472 Dependent Mean 61.60603 Adj R-Sq 0.5432 Coeff Var 4.42741 Parameter Estimates Parameter Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 62.39929 0.26211 238.06 <.0001 centage 1 2.03130 0.17307 11.74 <.0001     PAGE  PAGE 6 P Q R ) * E e    ;fM1h/sh/sCJOJQJ^JaJfHq 1h@IhCJOJQJ^JaJfHq hhfHq h[fHq h/sfHq hFfHq hfHq h0h0fHq h05>*h@IfHq h0fHq hMh3\hMh3\5CJ$aJ$Q R   4 5 f g  M 7$8$H$gd/sgd/s 7$8$H$gd0gd@Igd0$a$gd3\  5 < з|fKKK5+h0CJOJQJ^JaJfHq 4h@Ih3\CJOJQJ\^JaJfHq +h@ICJOJQJ^JaJfHq hMh3\CJOJQJ^JaJh/sCJOJQJ^JaJh0CJOJQJ^JaJh@ICJOJQJ^JaJ1h@Ih3\CJOJQJ^JaJfHq 1h@Ih@ICJOJQJ^JaJfHq +h/sCJOJQJ^JaJfHq  2"#$%bc.pP 7$8$H$gd4WgdV8gd3\ 7$8$H$gd3\ 7$8$H$gd/sOPTUWȲ~~~pbTF?5hPhP5>* hP5>*hPCJOJQJ^JaJh3\CJOJQJ^JaJh<CJOJQJ^JaJhCJOJQJ^JaJ4h<h3\CJOJQJ\^JaJfHq 1h<h3\CJOJQJ^JaJfHq +hCJOJQJ^JaJfHq hMh3\CJOJQJ^JaJ1hV8h4WCJOJQJ^JaJfHq h4WCJOJQJ^JaJlUVW!Z[Ggd3\gd< 7$8$H$gd< 7$8$H$gd3\gd4W 7$8$H$gd4Wlmn0LQRS 7$8$H$gdE2 7$8$H$gdV8 7$8$H$gd3\gd3\ 7$8$H$gdP(@Nijkln-ů{{{eOe+h!CJOJQJ^JaJfHq +hV8CJOJQJ^JaJfHq 4h<h3\CJOJQJ\^JaJfHq 1h<h3\CJOJQJ^JaJfHq +h.YCJOJQJ^JaJfHq h.Yh3\h!hV8h0h[h.Y hPhPhPhPCJOJQJ^JaJ hP5>*-ILOQRShpͲ͜znbnN7-hE2hE2CJOJQJaJfHq 'hV8CJOJQJaJfHq h4WfHq hE2fHq h!fHq +hV8CJOJQJ^JaJfHq +h3\CJOJQJ^JaJfHq 4h<h3\CJOJQJ\^JaJfHq 1h<h3\CJOJQJ^JaJfHq 1h<hV8CJOJQJ^JaJfHq 5HjprstlmVWyz 7BVq 7$8$H$gd< 7$8$H$gd\gd3\$a$gdE2 7$8$H$gdE2pqrstklmUW̶vh`RNFNB:B6Nhe0he0ht5hthe0h!5h!hPCJOJQJ^JaJhPhP5h\CJOJQJ^JaJhpXh\fHq h!fHq hE2fHq 1h4WhpXCJOJQJ^JaJfHq +hE2CJOJQJ^JaJfHq +h!CJOJQJ^JaJfHq :jh4WhE2CJOJQJU^JaJfHq Wz{ $(/57:;ABUʹjYKh<CJOJQJ^JaJ h!h3\CJOJQJ^JaJ+htCJOJQJ^JaJfHq 1h!h<CJOJQJ^JaJfHq 4h!h\CJOJQJ\^JaJfHq 1h!h\CJOJQJ^JaJfHq h!h#jVh^h^EHUjg[K h^CJUVaJh^jh^Uh#UV567<=NPqrst!! 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