Heteroscedasticity
Source:../../../../R/Packages/statistics/olsrr/vignettes/heteroskedasticity.Rmd
heteroskedasticity.RmdIntroduction
One of the assumptions made about residuals/errors in OLS regression is that the errors have the same but unknown variance. This is known as constant variance or homoscedasticity. When this assumption is violated, the problem is known as heteroscedasticity.
Consequences of Heteroscedasticity
- The OLS estimators and regression predictions based on them remains unbiased and consistent.
- The OLS estimators are no longer the BLUE (Best Linear Unbiased Estimators) because they are no longer efficient, so the regression predictions will be inefficient too.
- Because of the inconsistency of the covariance matrix of the estimated regression coefficients, the tests of hypotheses, (t-test, F-test) are no longer valid.
olsrr provides the following 4 tests for detecting heteroscedasticity:
- Bartlett Test
- Breusch Pagan Test
- Score Test
- F Test
Bartlett Test
Bartlett’s test is used to test if variances across samples is equal. It is sensitive to departures from normality. The Levene test is an alternative test that is less sensitive to departures from normality.
You can perform the test using 2 continuous variables, one continuous and one grouping variable, a formula or a linear model.
Use grouping variable
ols_test_bartlett(hsb, 'read', group_var = 'female')##
## Bartlett's Test of Homogenity of Variances
## ------------------------------------------------
## Ho: Variances are equal across groups
## Ha: Variances are unequal for atleast two groups
##
## Test Summary
## ----------------------------
## DF = 1
## Chi2 = 0.1866579
## Prob > Chi2 = 0.6657129Using variables
ols_test_bartlett(hsb, 'read', 'write')##
## Bartlett's Test of Homogenity of Variances
## ------------------------------------------------
## Ho: Variances are equal across groups
## Ha: Variances are unequal for atleast two groups
##
## Data
## ---------------------
## Variables: read write
##
## Test Summary
## ----------------------------
## DF = 1
## Chi2 = 1.222871
## Prob > Chi2 = 0.2687979Breusch Pagan Test
Breusch Pagan Test was introduced by Trevor Breusch and Adrian Pagan in 1979. It is used to test for heteroskedasticity in a linear regression model and assumes that the error terms are normally distributed. It tests whether the variance of the errors from a regression is dependent on the values of the independent variables. It is a \(\chi^{2}\) test.
You can perform the test using the fitted values of the model, the predictors in the model and a subset of the independent variables. It includes options to perform multiple tests and p value adjustments. The options for p value adjustments include Bonferroni, Sidak and Holm’s method.
Use fitted values of the model
model <- lm(mpg ~ disp + hp + wt + drat, data = mtcars)
ols_test_breusch_pagan(model)##
## Breusch Pagan Test for Heteroskedasticity
## -----------------------------------------
## Ho: the variance is constant
## Ha: the variance is not constant
##
## Data
## -------------------------------
## Response : mpg
## Variables: fitted values of mpg
##
## Test Summary
## ---------------------------
## DF = 1
## Chi2 = 1.429672
## Prob > Chi2 = 0.231818Use independent variables of the model
model <- lm(mpg ~ disp + hp + wt + drat, data = mtcars)
ols_test_breusch_pagan(model, rhs = TRUE)##
## Breusch Pagan Test for Heteroskedasticity
## -----------------------------------------
## Ho: the variance is constant
## Ha: the variance is not constant
##
## Data
## --------------------------
## Response : mpg
## Variables: disp hp wt drat
##
## Test Summary
## ----------------------------
## DF = 4
## Chi2 = 1.513808
## Prob > Chi2 = 0.8241927Use independent variables of the model and perform multiple tests
model <- lm(mpg ~ disp + hp + wt + drat, data = mtcars)
ols_test_breusch_pagan(model, rhs = TRUE, multiple = TRUE)##
## Breusch Pagan Test for Heteroskedasticity
## -----------------------------------------
## Ho: the variance is constant
## Ha: the variance is not constant
##
## Data
## --------------------------
## Response : mpg
## Variables: disp hp wt drat
##
## Test Summary (Unadjusted p values)
## ----------------------------------------------
## Variable chi2 df p
## ----------------------------------------------
## disp 1.2355345 1 0.2663334
## hp 0.9209878 1 0.3372157
## wt 1.2529988 1 0.2629805
## drat 1.1668486 1 0.2800497
## ----------------------------------------------
## simultaneous 1.5138083 4 0.8241927
## ----------------------------------------------Bonferroni p value Adjustment
model <- lm(mpg ~ disp + hp + wt + drat, data = mtcars)
ols_test_breusch_pagan(model, rhs = TRUE, multiple = TRUE, p.adj = 'bonferroni')##
## Breusch Pagan Test for Heteroskedasticity
## -----------------------------------------
## Ho: the variance is constant
## Ha: the variance is not constant
##
## Data
## --------------------------
## Response : mpg
## Variables: disp hp wt drat
##
## Test Summary (Bonferroni p values)
## ----------------------------------------------
## Variable chi2 df p
## ----------------------------------------------
## disp 1.2355345 1 1.0000000
## hp 0.9209878 1 1.0000000
## wt 1.2529988 1 1.0000000
## drat 1.1668486 1 1.0000000
## ----------------------------------------------
## simultaneous 1.5138083 4 0.8241927
## ----------------------------------------------Sidak p value Adjustment
model <- lm(mpg ~ disp + hp + wt + drat, data = mtcars)
ols_test_breusch_pagan(model, rhs = TRUE, multiple = TRUE, p.adj = 'sidak')##
## Breusch Pagan Test for Heteroskedasticity
## -----------------------------------------
## Ho: the variance is constant
## Ha: the variance is not constant
##
## Data
## --------------------------
## Response : mpg
## Variables: disp hp wt drat
##
## Test Summary (Sidak p values)
## ----------------------------------------------
## Variable chi2 df p
## ----------------------------------------------
## disp 1.2355345 1 0.7102690
## hp 0.9209878 1 0.8070305
## wt 1.2529988 1 0.7049362
## drat 1.1668486 1 0.7313356
## ----------------------------------------------
## simultaneous 1.5138083 4 0.8241927
## ----------------------------------------------Holm’s p value Adjustment
model <- lm(mpg ~ disp + hp + wt + drat, data = mtcars)
ols_test_breusch_pagan(model, rhs = TRUE, multiple = TRUE, p.adj = 'holm')##
## Breusch Pagan Test for Heteroskedasticity
## -----------------------------------------
## Ho: the variance is constant
## Ha: the variance is not constant
##
## Data
## --------------------------
## Response : mpg
## Variables: disp hp wt drat
##
## Test Summary (Holm's p values)
## ----------------------------------------------
## Variable chi2 df p
## ----------------------------------------------
## disp 1.2355345 1 0.7990002
## hp 0.9209878 1 0.3372157
## wt 1.2529988 1 1.0000000
## drat 1.1668486 1 0.5600994
## ----------------------------------------------
## simultaneous 1.5138083 4 0.8241927
## ----------------------------------------------Score Test
Test for heteroskedasticity under the assumption that the errors are independent and identically distributed (i.i.d.). You can perform the test using the fitted values of the model, the predictors in the model and a subset of the independent variables.
Use fitted values of the model
model <- lm(mpg ~ disp + hp + wt + qsec, data = mtcars)
ols_test_score(model)##
## Score Test for Heteroskedasticity
## ---------------------------------
## Ho: Variance is homogenous
## Ha: Variance is not homogenous
##
## Variables: fitted values of mpg
##
## Test Summary
## ----------------------------
## DF = 1
## Chi2 = 0.5163959
## Prob > Chi2 = 0.4723832Use independent variables of the model
model <- lm(mpg ~ disp + hp + wt + qsec, data = mtcars)
ols_test_score(model, rhs = TRUE)##
## Score Test for Heteroskedasticity
## ---------------------------------
## Ho: Variance is homogenous
## Ha: Variance is not homogenous
##
## Variables: disp hp wt qsec
##
## Test Summary
## ----------------------------
## DF = 4
## Chi2 = 2.039404
## Prob > Chi2 = 0.7285114Specify variables
model <- lm(mpg ~ disp + hp + wt + qsec, data = mtcars)
ols_test_score(model, vars = c('disp', 'hp'))##
## Score Test for Heteroskedasticity
## ---------------------------------
## Ho: Variance is homogenous
## Ha: Variance is not homogenous
##
## Variables: disp hp
##
## Test Summary
## ----------------------------
## DF = 2
## Chi2 = 0.9983196
## Prob > Chi2 = 0.6070405F Test
F Test for heteroskedasticity under the assumption that the errors are independent and identically distributed (i.i.d.). You can perform the test using the fitted values of the model, the predictors in the model and a subset of the independent variables.
Use fitted values of the model
model <- lm(mpg ~ disp + hp + wt + qsec, data = mtcars)
ols_test_f(model)##
## F Test for Heteroskedasticity
## -----------------------------
## Ho: Variance is homogenous
## Ha: Variance is not homogenous
##
## Variables: fitted values of mpg
##
## Test Summary
## -------------------------
## Num DF = 1
## Den DF = 30
## F = 0.4920617
## Prob > F = 0.4884154Use independent variables of the model
model <- lm(mpg ~ disp + hp + wt + qsec, data = mtcars)
ols_test_f(model, rhs = TRUE)##
## F Test for Heteroskedasticity
## -----------------------------
## Ho: Variance is homogenous
## Ha: Variance is not homogenous
##
## Variables: disp hp wt qsec
##
## Test Summary
## -------------------------
## Num DF = 4
## Den DF = 27
## F = 0.4594694
## Prob > F = 0.7647271Specify variables
model <- lm(mpg ~ disp + hp + wt + qsec, data = mtcars)
ols_test_f(model, vars = c('disp', 'hp'))##
## F Test for Heteroskedasticity
## -----------------------------
## Ho: Variance is homogenous
## Ha: Variance is not homogenous
##
## Variables: disp hp
##
## Test Summary
## -------------------------
## Num DF = 2
## Den DF = 29
## F = 0.4669306
## Prob > F = 0.631555