9.4 - Equal Slopes Model: Salary Example

Using Technology

Using our Salary example and the data in the table below, we can run through the steps for the ANCOVA.

Females		Males
Salary	Years	Salary	Years
80	5	78	3
50	3	43	1
30	2	103	5
20	1	48	2
60	4	80	4

Step 1: Are all regression slopes = 0?

A simple linear regression can be run for each treatment group, Males and Females. Running these procedures using statistical software we get the following:

Males

Use the following SAS code:

data equal_slopes;
input gender $ salary years;
datalines;
m 78 3
m 43 1
m 103 5
m 48 2
m 80 4
f 80 5
f 50 3
f 30 2
f 20 1
f 60 4
;
proc reg data=equal_slopes;
where gender='m';
model salary=years;
title 'Males';
run; quit;

And here is the output that you get:

The REG Procedure Mode1:: MODEL1 Dependent Variable: salary
Number of Observations Read	5
Number of Observations Used	5

Analysis of Variance
Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
Model	1	2310.40000	2310.40000	44.78	0.0068
Error	3	154.80000	51.60000
Corrected Total	4	2465.20000

Females

Use the following SAS code:

data equal_slopes;
input gender $ salary years;
datalines;
m 78 3
m 43 1
m 103 5
m 48 2
m 80 4
f 80 5
f 50 3
f 30 2
f 20 1
f 60 4
;
proc reg data=equal_slopes;
where gender='f';
model salary=years;
title 'Females';
run; quit;

And here is the output for this run:

The REG Procedure Mode1:: MODEL1 Dependent Variable: salary
Number of Observations Read	5
Number of Observations Used	5

Analysis of Variance
Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
Model	1	2250.00000	2250.00000	225.00	0.0006
Error	3	30.00000	10.00000
Corrected Total	4	2280.00000

In both cases, the simple linear regressions are significant so the slopes are not = 0.

Step 2: Are the slopes equal?

We can test for this using our statistical software.

In SAS we now use proc mixed and include the covariate in the model.

We will also include a ‘treatment × covariate’ interaction term and the significance of this term answers our question. If the slopes differ significantly among treatment levels, the interaction p-value will be < 0.05.

data equal_slopes;
input gender $ salary years;
datalines;
m 78 3
m 43 1
m 103 5
m 48 2
m 80 4
f 80 5
f 50 3
f 30 2
f 20 1
f 60 4
;
proc mixed data=equal_slopes method=type3;
class gender;
model salary = gender years gender*years;
run;

Note! In SAS, we specify the treatment in the class statement, indicating that these are categorical levels. By NOT including the covariate in the class statement, it will be treated as a continuous variable for regression in the model statement.

The Mixed Procedure Type 3 Tests of Fixed Effects
Effect	Num DF	Den DF	F Value	Pr > F
years	1	6	148.06	<.0001
gender	1	6	7.01	0.0381
years*gender	1	6	0.01	0.9384

Here we see that the slopes are equal and in a plot of the regressions, we see that the lines are parallel.

To obtain the plot in SAS, we can use the following SAS code:

ods graphics on;
proc sgplot data=equal_slopes;
styleattrs datalinepatterns=(solid);
reg y=salary x=years / group=gender;
run;

Step 3: Fit an Equal Slopes Model

We can now proceed to fit an Equal Slopes model by removing the interaction term. Again, we will use our statistical software SAS.

data equal_slopes;
input gender $ salary years;
datalines;
m 78 3
m 43 1
m 103 5
m 48 2
m 80 4
f 80 5
f 50 3
f 30 2
f 20 1
f 60 4
;
proc mixed data=equal_slopes method=type3;
class gender;
model salary = gender years;
lsmeans gender / pdiff adjust=tukey;
/* Tukey unnecessary with only two treatment levels */
title 'Equal Slopes Model';
run;

We obtain the following results:

The Mixed Procedure Type 3 Tests of Fixed Effects
Effect	Num DF	Den DF	F Value	Pr > F
years	1	7	172.55	<.0001
gender	1	7	47.46	0.0002

Least Squares Means
Effect	gender	Estimate	Standard Error	DF	t Value	Pr > \|t\|
gender	f	48.0000	2.2991	7	20.88	<.0001
gender	m	70.4000	2.2991	7	30.62	<.0001

Note! In SAS, the model statement automatically creates an intercept. This produces the correct table for testing the effects (using the adjusted sum of squares). However, including the intercept technically over-parameterizes the ANCOVA model resulting in an additional calculation step to obtain the regression equations. To get the intercepts for the covariate directly, we can re-parameterize the model by suppressing the intercept (noint) and then specifying that we want the solutions (solution) to the model. However, this should only be done to find the equation estimates, not to test effects.

Here is what the SAS code looks like for this:

data equal_slopes;
input gender $ salary years;
datalines;
m 78 3
m 43 1
m 103 5
m 48 2
m 80 4
f 80 5
f 50 3
f 30 2
f 20 1
f 60 4
;
proc mixed data=equal_slopes method=type3;
class gender;
model salary = gender years / noint solution;
ods select SolutionF;
title 'Equal Slopes Model';
run;

Here is the output:

Solution for Fixed Effects
Effect	gender	Estimate	Standard Error	DF	t Value	Pr > \|t\|
gender	f	2.7000	4.1447	7	0.65	0.5356
gender	m	25.1000	4.1447	7	6.06	0.0005
years		15.1000	1.1495	7	13.14	<.0001

In the first section of the output above a separate intercept is reported for each gender, the ‘Estimate’ value for each gender, and a common slope for both genders, labeled ‘Years’.

Thus, the estimated regression equation for Females is \(\hat{y} = 2.7 + 15.1 \text{(Years)}\), and for males it is \(\hat{y} = 25.1 + 15.1 \text{(Years)}\)

To this point in this analysis, we can see that 'gender' is now significant. By removing the impact of the covariate, we went from

Type 3 Tests of Fixed Effects
Effect	Num DF	Den DF	F Value	Pr > F
gender	1	8	2.11	0.1840

(without covariate consideration)

gender	1	7	47.46	0.0002

(adjusting for the covariate)

Using our Salary example and the data in the table below, we can run through the steps for the ANCOVA. On this page, we will go through the steps using Minitab.

Females		Males
Salary	Years	Salary	Years
80	5	78	3
50	3	43	1
30	2	103	5
20	1	48	2
60	4	80	4

Step 1: Are all regression slopes = 0?

A simple linear regression can be run for each treatment group, Males and Females. To perform regression analysis on each gender group in Minitab, we will have to subdivide the salary data manually and separately, saving the male data into the Male Salary Dataset and the female data into the Female Salary dataset.

Running these procedures using statistical software we get the following:

Males

Open the Male dataset in the Minitab project file (Male Salary Dataset).

Then, from the menu bar, select Stat > Regression > Regression > Fit Regression Model

In the pop-up window, select salary into Response and years into Predictors as shown below.

Minitab Regression window with Responses=salary, Continuous predictors=years

Click OK, and Minitab will output the following.

Regression Analysis: Salary versus years

Regression Equation

salary = 24.8 + 15.2 years

Coefficients

Term	Coef	SE Coef	T-Value	P-Value	VIF
Constant	24.80	7.53	3.29	0.046
years	15.20	2.27	6.69	0.007	1.00

Model Summary

S	R-sq	R-sq(adj)	R-sq(pred)
7.18331	R-Sq = 93.7%	91.6%	85.94%

Analysis of Variance

Source	DF	SS	MS	F-Value	P-Value
Regression	1	2310.4	2310.40	44.78	0.007
years	1	2310.4	2310.40	44.78	0.007
Residual Error	3	154.8	51.6
Total	4	2465.2

Females

Open Minitab dataset Female Salary Dataset. Follow the same procedure as was done for the Male dataset and Minitab will output the following:

Regression Analysis: Salary versus years

Regression Equation

salary = 3.00 + 15.00 years

Coefficients

Term	Coef	SE Coef	T-Value	P-Value	VIF
Constant	3.00	3.32	0.90	0.432
years	15.00	1.00	15.00	0.001	1.00

Model Summary

S	R-sq	R-sq(adj)	R-sq(pred)
3.16228	98.68%	98.25%	95.92%

Analysis of Variance

Source	DF	SS	MS	F-Value	P-Value
Regression	1	2250.0	2250.0	225.00	0.001
years	1	2250.0	2250.0	225.00	0.001
Residual Error	3	30.0	10.0
Total	4	2280.0

In both cases, the simple linear regressions are significant, so the slopes are not = 0.

Step 2: Are the slopes equal?

We can test for this using our statistical software. In Minitab, we must now use GLM (general linear model) and be sure to include the covariate in the model. We will also include a ‘treatment x covariate’ interaction term and the significance of this term is what answers our question. If the slopes differ significantly among treatment levels, the interaction p-value will be < 0.05.

First, open the dataset in the Minitab project file Salary Dataset. Then, from the menu select Stat > ANOVA > General Linear Model > Fit General Linear Model

In the dialog box, select salary into Responses, gender into Factors, and years into Covariates.

Minitab General Linear Model window with Responses= salary, Factors=gender and Covariates=years

To add the interaction term, first click Model…. Then, use the shift key to highlight gender and years, and click Add. Click OK, then OK again, and Minitab will display the following output.

Analysis of Variance

Source	DF	Adj SS	Adj MS	F-Value	P-Value
year	1	4560.20	4560.20	148.06	0.000
gender	1	216.02	216.02	7.01	0.038
years*gender	1	0.20	0.20	0.01	0.938
Error	6	184.80	30.80
Total	9	5999.60

It is clear the interaction term is not significant. This suggests the slopes are equal. In a plot of the regressions, we can also see that the lines are parallel.

Step 3: Fit an Equal Slopes Model

We can now proceed to fit an Equal Slopes model by removing the interaction term. This can be easily accomplished by starting again with STAT > ANOVA > General Linear Model > Fit General Linear Model

Click OK, then OK again, and Minitab will display the following output.

Analysis of Variance

Source	DF	Adj SS	Adj MS	F-Value	P-Value
year	1	4560.20	4560.20	172.55	0.000
gender	1	1254.4	1254.40	47.46	0.000
Error	7	185.0	26.43
Total	9	5999.6

To generate the mean comparisons select STAT > ANOVA > General Linear Model > Comparisons... and fill in the dialog box as seen below.

Minitab Comparisions windo with Response=salary, Type of Comparison=Pairwise and Method= Tukey

Click OK and Minitab will produce the following output.

Comparison of salary

Tukey Pairwise Comparisons: gender

Grouping information Using the Tukey Method and 95% Confidence

gender N Mean Grouping
Male 5 70.4 A
gender 5 48.0 B
Means that do not share a letter are significantly different.

gender	N	Mean	Grouping
Male	5	70.4	A
gender	5	48.0	B

First, we can input the data manually.

gender = c(rep("m",5),rep("f",5))
salary = c(78,43,103,48,80,80,50,30,20,60)
years = c(3,1,5,2,4,5,3,2,1,4)
salary_data = data.frame(salary,gender,years)

We can apply a simple linear regression for the male subset of the data and display the results using summary.

male_data = subset(salary_data,gender=="m")
lm_male = lm(salary~years,male_data)
summary(lm_male)

Call:
lm(formula = salary ~ years, data = male_data)

Residuals:
   1    2    3    4    5 
 7.6  3.0  2.2 -7.2 -5.6 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)   24.800      7.534   3.292  0.04602 * 
years         15.200      2.272   6.691  0.00681 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 7.183 on 3 degrees of freedom
Multiple R-squared:  0.9372,	Adjusted R-squared:  0.9163 
F-statistic: 44.78 on 1 and 3 DF,  p-value: 0.006809

Next, we apply a simple linear regression for the female subset.

female_data = subset(salary_data,gender=="f")
lm_female = lm(salary~years,female_data)
summary(lm_female)

Call:
lm(formula = salary ~ years, data = female_data)

Residuals:
 6  7  8  9 10 
 2  2 -3  2 -3 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)    3.000      3.317   0.905 0.432389    
years         15.000      1.000  15.000 0.000643 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.162 on 3 degrees of freedom
Multiple R-squared:  0.9868,	Adjusted R-squared:  0.9825 
F-statistic:   225 on 1 and 3 DF,  p-value: 0.0006431

It is clear the regression for both treatments is significant. We continue to test for unequal slopes in the full dataset using an interaction term.

options(contrasts=c("contr.sum","contr.poly"))
lm_unequal = lm(salary~gender+years+gender:years,salary_data)
aov3_unequal = car::Anova(lm_unequal,type=3)
aov3_unequal

Anova Table (Type III tests)

Response: salary
             Sum Sq Df  F value    Pr(>F)    
(Intercept)   351.3  1  11.4055   0.01491 *  
gender        216.0  1   7.0136   0.03811 *  
years        4560.2  1 148.0584 1.874e-05 ***
gender:years    0.2  1   0.0065   0.93839    
Residuals     184.8  6                       
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The interaction term is not significant, suggesting the slopes do not differ significantly. We simplify the model to the equal-slope model without an interaction term.

lm_equal = lm(salary~gender+years,salary_data)
aov3_equal = car::Anova(lm_equal,type=3)
aov3_equal

Anova Table (Type III tests)

Response: salary
            Sum Sq Df F value    Pr(>F)    
(Intercept)  351.3  1  13.292 0.0082232 ** 
gender      1254.4  1  47.464 0.0002335 ***
years       4560.2  1 172.548 3.458e-06 ***
Residuals    185.0  7                      
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

This is the final model since all terms are significant. We can then produce the LS means for the gender levels.

aov1_equal = aov(lm_equal)
lsmeans_gender = emmeans::emmeans(aov1_equal,~gender) 
lsmeans_gender

 gender emmean  SE df lower.CL upper.CL
 f        48.0 2.3  7     42.6     53.4
 m        70.4 2.3  7     65.0     75.8

Confidence level used: 0.95

We can also find the regression equation coefficients. Note the female level was used as the reference level by default.

  aov1_equal$coefficients

(Intercept)     genderm       years 
        2.7        22.4        15.1

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9.4 - Equal Slopes Model: Salary Example

Using Technology

Step 1: Are all regression slopes = 0?

Males

Females

Step 2: Are the slopes equal?

Step 3: Fit an Equal Slopes Model

Step 1: Are all regression slopes = 0?

Males

Regression Analysis: Salary versus years

Regression Equation

Coefficients

Model Summary

Analysis of Variance

Females

Regression Analysis: Salary versus years

Regression Equation

Coefficients

Model Summary

Analysis of Variance

Step 2: Are the slopes equal?

Analysis of Variance

Step 3: Fit an Equal Slopes Model

Analysis of Variance

Comparison of salary

Tukey Pairwise Comparisons: gender

Grouping information Using the Tukey Method and 95% Confidence