7.4.4. Regression with Replicates
This is a test of linearity for bivariate regressions when the data contains multiple measurements of the dependent variable for each value of the independent variable. The null hypothesis tested is “population regression is linear” against the alternative hypothesis “population regression is not linear”.
Select at least one variable as [Factor] and at least one data variable as [Dependent]. The procedure is run separately for each [Factor] / [Dependent] pair. Output consists of simple regression results, and ANOVA of regression table (testing the null hypothesis that “the slope of the regression line is zero”), another ANOVA table where the among groups variation is broken down into the Linear Regression and its error sum of squares. The test statistic is the Ftest for regression error term, which is defined as:
Example 1
Examples 17.8a and 17.8b on pp. 349, 350 from Zar, J. H. (2010). The null hypothesis that “the population regression is linear” is tested at a 95% confidence level.
The table format given in the book can be transformed into the factor format by using UNISTAT’s Data → Stack Columns procedure and the Level() function (see 3.4.2.5. Statistical Functions). All data on systolic blood pressure should be stacked in a column Pressure (the Y variable) and all data on ages (the X variable) should be expanded to form a column Age to keep track of the age groups of pressure measurements.
Open ANOTESTS, select Statistics 1 → Tests for ANOVA → Regression with Replicates and select Age (C5) as [Factor] and Pressure (C6) as [Dependent] to obtain the following results:
Regression with Replicates
Regression results
Constant = 
68.7849 
Slope = 
1.3031 
Rsquared = 
0.9827 
Standard Error = 
2.5702 
ANOVA of regression
Due To 
Sum of Squares 
DoF 
Mean Square 
Fstat 
Prob 
Regression 
6750.289 
1 
6750.289 
1021.819 
0.0000 
Error 
118.911 
18 
6.606 


Total 
6869.200 
19 
361.537 


Test of linearity
Due To 
Sum of Squares 
DoF 
Mean Square 
Fstat 
Prob 
Among groups 
6751.933 
4 
1687.983 
215.916 
0.0000 
Regression 
6750.289 
1 
6750.289 
863.454 
0.0000 
Error 
1.644 
3 
0.548 
0.070 
0.9750 
Within groups 
117.267 
15 
7.818 


Total 
6869.200 
19 
361.537 


Since the probability value for Regression in the ANOVA of regression table is less than 5% reject the null hypothesis that “the regression slope is zero”. The test of linearity is the Fstatistic on regression error, which is 0.070 with a 97.5% probability, therefore do not reject the null hypothesis of linearity.
Example 2
Example 11.2 on p. 316 in Armitage & Berry (2002). Data on radiographic assessments of bone healing for three doses of vitamin D are given.
The format of Table 9.3 in the book is not suitable for analysis in UNISTAT. All data should be stacked in a single column Radiography and a factor column Dose created to keep track of the group memberships.
Open ANOTESTS, select Statistics 1 → Tests for ANOVA → Regression with Replicates and select Dose (C7) as [Factor] and Radiography (C8) as [Dependent] to obtain the following results:
Regression with Replicates
For Radiography, classified by Dose
Constant = 
1.2195 
Slope = 
0.7876 
Rsquared = 
0.2399 
Standard Error = 
1.2408 
ANOVA of regression
Due To 
Sum of Squares 
DoF 
Mean Square 
Fstat 
Prob 
Regression 
14.089 
1 
14.089 
9.151 
0.0052 
Error 
44.645 
29 
1.539 


Total 
58.734 
30 
1.958 


Test of linearity
Due To 
Sum of Squares 
DoF 
Mean Square 
Fstat 
Prob 
Among groups 
16.992 
2 
8.496 
5.699 
0.0084 
Regression 
14.089 
1 
14.089 
9.451 
0.0047 
Error 
2.903 
1 
2.903 
1.948 
0.1738 
Within groups 
41.742 
28 
1.491 


Total 
58.734 
30 
1.958 


The test of linearity is the Fstatistic on regression error, which is 1.948 with a 17% tail probability. Therefore, do not reject the null hypothesis of linearity.