Of interest is the slope of the regression of weight on height in a specific population of students. For this example two hypotheses are tested with (a) and (b) . A dataset of measurements on a random sample of students has been used (dataset in Table A.6).

SAS code

* Simple linear regression including test for H0: beta_1=0;
proc reg data=students;
 model weight=height;
run;
* Perform test for H0: beta_1=0.5;
proc reg data=students;
 model weight=height;
 test height=0.5;
run;
quit;

                   Parameter Estimates
                  Parameter   Standard
Variable     DF   Estimate      Error    t Value   Pr> |t|
Intercept     1   -51.81816   35.76340   -1.45      0.1646
height        1     0.67892    0.20645    3.29      0.0041
                  The REG Procedure
                    Model: MODEL1
     Test 1 Results for Dependent Variable weight
                                Mean
Source             DF         Square    F Value    Pr> F
Numerator           1       94.54374       0.75    0.3975
Denominator        18      125.87535

• The SAS procedure proc reg is the standard procedure for linear regression. It is a powerful procedure and we use here only a tiny part of it.
• For the standard hypothesis the model dependent variable=independent variable statement is sufficient.
• For testing a special hypothesis you must add the test variable= value statement. Note, here a F-test is used, which is equivalent to the proposed t-test, because a squared t-distributed random variable with degrees of freedom is -distributed. The p-value stays the same. To get the t-test use the restrict variable= value statement.
• The quit; statement is used to terminate the procedure; proc reg is an interactive procedure and SAS then knows not to expect any further input.
• The p-values for the other hypothesis must be calculated by hand. For instance, for the p-value for hypothesis (B) is 1-probt(3.29,18)=0.0020 and for hypothesis (C) probt(3.29,18)=0.9980.

R code

# Read the data
y<-students$weight
x<-students$height
# Simple linear regression including test for H0: beta_1=0
reg<-summary(lm(y~x))
# Perform test for H0: beta_1=0.5
# Get estimated coefficient
beta_1<-reg$coeff[2,1]
# Get standard deviation of estimated coefficient
std_beta_1<-reg$coeff[“x”,2]
# Perform the test
t_value<-(beta_1-0.5)/std_beta_1
# Calculate p-value
p_value<-2*pt(-abs(t_value),18)
# Output result
# Simple linear regression
reg
# For hypothesis H0: beta_1=0.5
t_value
p_value

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept) -51.8182    35.7634  -1.449  0.16456
x             0.6789     0.2065   3.288  0.00408 **
---
Signif. codes:  0 *** 0.001 ** 0.01 * 0.05 . 0.1   1
> # For hypothesis H0: beta_1=0.5
> t_value
[1] 0.8666546
> p_value
[1] 0.3975371

• The function lm() performs a linear regression in R. The response variable is placed on the left-hand side of the symbol and the regressor variable on the right-hand side.
• The summary function gets R to return the estimates, p-values, etc. Here we store the values in the object reg.
• The standard hypothesis is performed by the function lm(). The hypothesis with is not covered by this function but it provides the necessary statistics which we store in above example code in the object reg. In the second part of the example we extract the estimated coefficient with the command reg$coeff[2,1] and its estimated standard deviations with the command reg$coeff[2,2]. These values are then used to perform the test.
• The p-values for the other hypothesis must be calculated by hand. For instance for the p-value for hypothesis (B) is 1-pt(3.29,18)=0.0020 and for hypothesis (C) pt(3.29,18)=0.9980.

Of interest is the intercept of the regression of weight on height in a specific population of students. For this example two hypotheses are tested with (a) and (b) . A dataset of measurements on a random sample of students has been used (dataset in Table A.6).

SAS code

* Simple linear regression including test for H0: beta_1=0;
proc reg data=students;
 model weight=height;
run;
* Perform test for H0: beta_0=10;
proc reg data=students;
 model weight=height;
 test intercept=10;
run;

                   Parameter Estimates
                  Parameter   Standard
Variable     DF   Estimate      Error    t Value   Pr> |t|
Intercept     1   -51.81816   35.76340   -1.45      0.1646
height        1     0.67892    0.20645    3.29      0.0041
                   The REG Procedure
                     Model: MODEL1
          Test 1 Results for Dependent Variable weight
                                Mean
Source             DF         Square    F Value    Pr> F
Numerator           1      376.09299       2.99    0.1010
Denominator        18      125.87535

• The SAS procedure proc reg is the standard procedure for linear regression. It is a powerful procedure and we use here only a tiny part of it.
• For the standard hypothesis the model dependent variable=independent variable statement is sufficient.
• For testing a special hypothesis you must add the test intercept value statement. Note, here a F-test is used, which is equivalent to the proposed t-test, because a squared t-distributed random variable with degrees of freedom is -distributed. The p-value stays the same. To get the t-test use the restrict variable= value statement.
• The quit; statement is used to terminate the procedure; proc reg is an interactive procedure and SAS then knows not to expect any further input.
• The p-values for the other hypothesis must be calculate by hand. For instance for the p-value for hypothesis (B) is 1-probt(-51.82,18)=1 and for hypothesis (C) probt(-51.82,18)=0.

R code

# Read the data
y<-students$weight
x<-students$height
# Simple linear regression
reg<-summary(lm(y~x))
# Perform test for H0: beta_0=10
# Get estimated coefficient
beta_0<-reg$coeff[1,1]
# Get standard deviation of estimated coefficient
std_beta_0<-reg$coeff[1,2]
# Perform the test
t_value<-(beta_0-10)/std_beta_0
# Calculate p-Value
p_value<-2*pt(-abs(t_value),18)
# Output result
# Simple linear regression
reg
# For hypothesis H0: beta_0=10
t_value
p_value

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept) -51.8182    35.7634  -1.449  0.16456
x             0.6789     0.2065   3.288  0.00408 **
---
Signif. codes:  0 *** 0.001 ** 0.01 * 0.05 . 0.1   1
>
> # For hypothesis H0: beta_0=10
> t_value
[1] -1.728531
> p_value
[1] 0.1010077

• The function lm() performs a linear regression in R. The response variable is placed on the left-hand side of the symbol and the regressor variable on the right-hand side.
• The summary function gets R to return the estimates, p-values, etc. Here we store the values in the object reg.
• The standard hypothesis is performed by the function lm(). The hypothesis with is not covered by this function but it provides the necessary statistics which we store in the example code in the object reg. In the second part of the example we extract the estimated coefficient with the command reg$coeff[1,1] and its estimated standard deviation with the command reg$coeff[1,2]. These values are then used to perform the test.
• The p-values for the other hypothesis must be calculated by hand. For instance for the p-value for hypothesis (B) is 1-pt(-51.82,18)=1 and for hypothesis (C) pt(-51.82,18)=0.

Of interest is the effect of sex in a regression of weight on height and sex in a specific population of students. The variable sex needs to becoded as a dummy variable for the regression model. In our example we choose the outcome male as reference, hence the new variable sex takes the value for female students and for male students. We test the hypothesis . A dataset of measurements on a random sample of students has been used (dataset in Table A.6).

SAS code

* Create dummy variable for sex with reference male;
data reg;
 set students;
 if sex=1 then s=0;
 if sex=2 then s=1;
run;
* Perform linear regression;
proc reg data=reg;
 model weight=height s;
run;
quit;

                  Parameter   Standard
Variable     DF    Estimate      Error   t Value   Pr> |t|
Intercept     1   -44.10291   39.97051     -1.10     0.2852
height        1     0.64182    0.22489      2.85     0.0110
s             1    -2.60868    5.46554     -0.48     0.6392

• The SAS procedure proc reg is the standard procedure for linear regression. It is a powerful procedure and we use here only a tiny part of it.
• For the standard hypothesis the model dependent variable=independent variables statement is sufficient. The independent variables are separated by blanks.
• Categorical variables can also be regressors but care must be taken as to which value is the reference value. Here we code sex as the dummy variable, with males as the reference group.
• The quit; statement is used to terminate the procedure; proc reg is an interactive procedure and SAS then knows not to expect any further input.
• The p-values for the other hypothesis must be calculate by hand. For instance for the variable sex the p-value for hypothesis (B) is 1-probt(-0.48,18)= 0.6815 and for hypothesis (C) probt(-0.48,18)=0.3185.
• For testing a special hypothesis you must add the test variable= value statement. Note, here a F-test is used, which is equivalent to the proposed t-test, because a squared t-distributed random variable with degrees of freedom is -distributed. The p-value stays the same. To get the t-test use the restrict variable= value statement.

R code

# Read the data
weight<-students$weight
height<-students$height
sex<-students$sex
# Multiple linear regression
summary(lm(weight~height+factor(sex)))

Coefficients:
             Estimate Std. Error t value Pr(>|t|)
(Intercept)  -44.1029    39.9705  -1.103    0.285
height         0.6418     0.2249   2.854    0.011 *
factor(sex)2  -2.6087     5.4655  -0.477    0.639
---
Signif. codes:  0 *** 0.001 ** 0.01 * 0.05 . 0.1   1

• The function lm() performs a linear regression in R. The response variable is placed on the left-hand side of the symbol and the regressor variables on the right-hand side separated by a plus (+).
• Categorical variables can also be regressors, but care must be taken as to which value is the reference value. We use the factor() function to tell R that sex is a categorical variable. We see from the output factor(sex)2 that the effect is for females and therefore the males are the reference. To switch these, recode the values of males and females.
• The summary function gets R to return the estimates, p-values, etc.
• The standard hypothesis is performed by the function lm(). The hypothesis is not covered by this function but it provides the necessary statistics which can then be used. See Test 16.1.1 on how to do so.
• The p-values for the other hypothesis must be calculated by hand. For instance, for the p-value for hypothesis (B) is 1-pt(-0.477,18)=0.6805 and for hypothesis (C) pt(-0.47729,18)=0.3196.

Of interest is the regression of weight on height and sex in a specific population of students. We test for overall significance of regression, hence the hypothesis . A dataset of measurements on a random sample of students has been used (dataset in Table A.6).

SAS code

proc reg data=reg;
 model weight=height sex;
run;
quit;

                   Analysis of Variance
                          Sum of      Mean
Source           DF      Squares    Square  F Value  Pr> F
Model             2   1391.20481  695.60241    5.29  0.0164
Error            17   2235.79519  131.51736
Corrected Total  19   3627.00000

• The SAS procedure proc reg is the standard procedure for linear regression. It is a powerful procedure and we use here only a tiny part of it.
• Categorical variables can also be regressors, but care must be taken as to which value is the reference value. Here we code sex as the dummy variable, with male as the reference group.
• The quit; statement is used to terminate the procedure; proc reg is an interactive procedure and SAS then knows not to expect any further input.

R code

summary(lm(students$weight~students$height
                                    +factor(students$sex)))

F-statistic: 5.289 on 2 and 17 DF,  p-value: 0.01637

• The function lm() performs a linear regression in R. The response variable is placed on the left-hand side of the symbol and the regressor variables on the right-hand side separated by a plus (+).
• We use the R function factor() to tell R that sex is a categorical variable.
• The summary function gets R to return parameter estimates, p-values for the overall F-tests, p-values for tests on individual regression parameters, etc.