# Homoscedasticity

The assumption of homoscedasticity (meaning “same variance”) is central to linear regression models.  Homoscedasticity describes a situation in which the error term (that is, the “noise” or random disturbance in the relationship between the independent variables and the dependent variable) is the same across all values of the independent variables.  Heteroscedasticity (the violation of homoscedasticity) is present when the size of the error term differs across values of an independent variable.  The impact of violating the assumption of homoscedasticity is a matter of degree, increasing as heteroscedasticity increases.

A simple bivariate example can help to illustrate heteroscedasticity: Imagine we have data on family income and spending on luxury items.  Using bivariate regression, we use family income to predict luxury spending.  As expected, there is a strong, positive association between income and spending.  Upon examining the residuals we detect a problem – the residuals are very small for low values of family income (almost all families with low incomes don’t spend much on luxury items) while there is great variation in the size of the residuals for wealthier families (some families spend a great deal on luxury items while some are more moderate in their luxury spending).  This situation represents heteroscedasticity because the size of the error varies across values of the independent variable.  Examining a scatterplot of the residuals against the predicted values of the dependent variable would show a classic cone-shaped pattern of heteroscedasticity.

The problem that heteroscedasticity presents for regression models is simple.  Recall that ordinary least-squares (OLS) regression seeks to minimize residuals and in turn produce the smallest possible standard errors.  By definition, OLS regression gives equal weight to all observations, but when heteroscedasticity is present, the cases with larger disturbances have more “pull” than other observations.  In this case, weighted least squares regression would be more appropriate, as it down-weights those observations with larger disturbances. ### Discover How We Assist to Edit Your Dissertation Chapters

Aligning theoretical framework, gathering articles, synthesizing gaps, articulating a clear methodology and data plan, and writing about the theoretical and practical implications of your research are part of our comprehensive dissertation editing services.

• Bring dissertation editing expertise to chapters 1-5 in timely manner.
• Track all changes, then work with you to bring about scholarly writing.
• Ongoing support to address committee feedback, reducing revisions.

A more serious problem associated with heteroscedasticity is the fact that the standard errors are biased.  Because the standard error is central to conducting significance tests and calculating confidence intervals, biased standard errors lead to incorrect conclusions about the significance of the regression coefficients.  Many statistical programs provide an option of robust standard errors to correct this bias.  Weighted least squares regression also addresses this concern but requires a number of additional assumptions.  Another approach for dealing with heteroscedasticity is to transform the dependent variable using one of the variance stabilizing transformations.  A logarithmic transformation can be applied to highly skewed variables, while count variables can be transformed using a square root transformation.  Overall however, the violation of the homoscedasticity assumption must be quite severe in order to present a major problem given the robust nature of OLS regression.

Statistics Solutions can assist with your quantitative analysis by assisting you to develop your methodology and results chapters. The services that we offer include:

Data Analysis Plan

Edit your research questions and null/alternative hypotheses

Write your data analysis plan; specify specific statistics to address the research questions, the assumptions of the statistics, and justify why they are the appropriate statistics; provide references

Justify your sample size/power analysis, provide references

Explain your data analysis plan to you so you are comfortable and confident

Two hours of additional support with your statistician

Quantitative Results Section (Descriptive Statistics, Bivariate and Multivariate Analyses, Structural Equation Modeling, Path analysis, HLM, Cluster Analysis)

Clean and code dataset

Conduct descriptive statistics (i.e., mean, standard deviation, frequency and percent, as appropriate)

Conduct analyses to examine each of your research questions

Write-up results

Provide APA 6th edition tables and figures

Explain chapter 4 findings

Ongoing support for entire results chapter statistics

Please call 727-442-4290 to request a quote based on the specifics of your research, schedule using the calendar on this page, or email [email protected]