Interpreting OLS results

Output generated from the OLS Regression tool includes the following:

• Output feature class

  Map of OLS Residuals

• Message window report of statistical results

  OLS Statistical Report

• Optional table of explanatory variable coefficients

  Coefficient table of OLS Model Results

• Optional table of regression diagnostics

  OLS Model Diagnostics Table

Each of these outputs is shown and described below as a series of steps for running OLS regression and interpreting OLS results.

(A) Open the Results window, if necessary. Then, run the OLS tool:

Ordinary Least Squares Tool Dialog Box

You will need to provide an Input Feature Class with a Unique ID Field, the Dependent Variable you want to model/explain/predict, and all the Explanatory Variables. You will also need to provide a path name for the Output Feature Class and, optionally, path names for the Coefficient Output Table and Diagnostic Output Table.

The primary output is the OLS summary report, which is written to the Results window. Right-clicking the Messages entry in the Results window and selecting View will display the OLS summary report in a Message dialog box. If you execute this tool in the foreground, the summary report will also be displayed in the progress dialog box.

(B) Examine the statistical report using the numbered steps described below:

Components of the OLS Statistical Report

Dissecting the Statistical Report

1. Assess model performance. Both the Multiple R-Squared and Adjusted R-Squared values are measures of model performance. Possible values range from 0.0 to 1.0. The Adjusted R-Squared value is always a bit lower than the Multiple R-Squared value, because it reflects model complexity (the number of variables) as it relates to the data and is consequently a more accurate measure of model performance. Adding an additional explanatory variable to the model will likely increase the Multiple R-Squared value but may decrease the Adjusted R-Squared value. Suppose you are creating a regression model of residential burglary (the number of residential burglaries associated with each census block is your dependent variable, y). An Adjusted R-Squared value of 0.84 would indicate that your model (your explanatory variables modeled using linear regression) explains approximately 84 percent of the variation in the dependent variable. Said another way, your model tells approximately 84% of the residential burglary "story."

   R-Squared Values Quantify Model Performance

2. Assess each explanatory variable in the model: Coefficient, Probability or Robust Probability, and Variance Inflation Factor (VIF). The coefficient for each explanatory variable reflects both the strength and type of relationship the explanatory variable has to the dependent variable. When the sign associated with the coefficient is negative, the relationship is negative (for example, the larger the distance from the urban core, the smaller the number of residential burglaries). When the sign is positive, the relationship is positive (for example, the larger the population, the larger the number of residential burglaries). Coefficients are given in the same units as their associated explanatory variables (a coefficient of 0.005 associated with a variable representing population counts may be interpreted as 0.005 people). The coefficient reflects the expected change in the dependent variable for every 1 unit change in the associated explanatory variable, holding all other variables constant (for example, a 0.005 increase in residential burglary is expected for each additional person in the census block, holding all other explanatory variables constant). The T test is used to assess whether or not an explanatory variable is statistically significant. The null hypothesis is that the coefficient is, for all intents and purposes, equal to zero (and consequently is NOT helping the model). When the probability or robust probability (p-values) is very small, the chance of the coefficient being essentially zero is also small. If the Koenker test (see below) is statistically significant, use the robust probabilities to assess explanatory variable statistical significance. Statistically significant probabilities have an asterisk (*) next to them. An explanatory variable associated with a statistically significant coefficient is important to the regression model if theory/common sense supports a valid relationship with the dependent variable, if the relationship being modeled is primarily linear, and if the variable is not redundant to any other explanatory variables in the model. The VIF measures redundancy among explanatory variables. As a rule of thumb, explanatory variables associated with VIF values larger than about 7.5 should be removed (one by one) from the regression model. If, for example, you have a population variable (the number of people) and an employment variable (the number of employed persons) in your regression model, you will likely find them to be associated with large VIF values indicating that both of these variables are telling the same "story;" one of them should be removed from your model.

   Assess which variables are statistically significant.

3. Assess model significance. Both the Joint F-Statistic and Joint Wald Statistic are measures of overall model statistical significance. The Joint F-Statistic is trustworthy only when the Koenker (BP) statistic (see below) is not statistically significant. If the Koenker (BP) statistic is significant, you should consult the Joint Wald Statistic to determine overall model significance. The null hypothesis for both of these tests is that the explanatory variables in the model are NOT effective. For a 95 percent confidence level, a p-value (probability) smaller than 0.05 indicates a statistically significant model.

   Assess the overall statistical significance of the regression model.

4. Assess Stationarity. The Koenker (BP) Statistic (Koenker's studentized Bruesch-Pagan statistic) is a test to determine whether the explanatory variables in the model have a consistent relationship to the dependent variable both in geographic space and in data space. When the model is consistent in geographic space, the spatial processes represented by the explanatory variables behave the same everywhere in the study area (the processes are stationary). When the model is consistent in data space, the variation in the relationship between predicted values and each explanatory variable does not change with changes in explanatory variable magnitudes (there is no heteroscedasticity in the model). Suppose you want to predict crime, and one of your explanatory variables is income. The model would have problematic heteroscedasticity if the predictions were more accurate for locations with small median incomes than they were for locations with large median incomes. The null hypothesis for this test is that the model is stationary. For a 95 percent confidence level, a p-value (probability) smaller than 0.05 indicates statistically significant heteroscedasticity and/or nonstationarity. When results from this test are statistically significant, consult the robust coefficient standard errors and probabilities to assess the effectiveness of each explanatory variable. Regression models with statistically significant nonstationarity are often good candidates for Geographically Weighted Regression (GWR) analysis.

   Assess stationarity: if the Koenker test is statistically significant (*), consult the robust probabilities to determine whether explanatory variable coefficients are significant.

5. Assess model bias. The Jarque-Bera statistic indicates whether or not the residuals (the observed/known dependent variable values minus the predicted/estimated values) are normally distributed. The null hypothesis for this test is that the residuals are normally distributed, so if you were to construct a histogram of those residuals, they would resemble the classic bell curve, or Gaussian distribution. When the p-value (probability) for this test is small (smaller than 0.05 for a 95 percent confidence level, for example), the residuals are not normally distributed, indicating your model is biased. If you also have statistically significant spatial autocorrelation of your residuals (see below), the bias may be the result of model misspecification (a key variable is missing from the model). Results from a misspecified OLS model are not trustworthy. A statistically significant Jarque-Bera test can also occur if you are trying to model non-linear relationships, if your data include influential outliers, or when there is strong heteroscedasticity (see above).

   Assess model bias.

6. Assess residual spatial autocorrelation. Always run the Spatial Autocorrelation (Moran's I) tool on the regression residuals to ensure that they are spatially random. Statistically significant clustering of high and/or low residuals (model under- and overpredictions) indicates a key variable is missing from the model (misspecification). OLS results cannot be trusted when the model is misspecified.

   Use the Spatial Autocorrelation tool to ensure that model residuals are not spatially autocorrelated.

7. Finally, review the section titled How Regression Models Go Bad in the Regression Analysis Basics document as a check that your OLS regression model is properly specified. Notice, too, that there is a section titled Notes on Interpretation at the end of the OLS statistical report to help you remember the purpose of each statistical test.

   The OLS report includes Notes to help you interpret diagnostic output.

(C) Examine Output Feature Class residuals. Over- and under-predictions for a properly specified regression model will be randomly distributed. Clustering of over- and/or under-predictions is evidence that you are missing at least one key explanatory variable. Examine the patterns in your model residuals to see if they provide clues about what those missing variables might be. Sometimes running Hot Spot Analysis on regression residuals will help you see the broader patterns in over- and under-predictions.

OLS Output: Mapped Residuals

(D) View the coefficient and diagnostic tables. Creating the coefficient and diagnostic tables is optional. While you are in the process of finding an effective model, you may elect not to create these tables. The model-building process is iterative, and you will likely try a large number of different models (different explanatory variables) until you settle on a few good ones. You can use the Corrected Akaike Information Criterion (AICc) on the report to compare different models. The model with the smaller AICc value is the better model (that is, taking into account model complexity, the model with the smaller AICc provides a better fit with the observed data). You should always create the coefficient and diagnostic tables for your final OLS models to capture the most important elements of the OLS report, including the list of explanatory variables used in the model with their coefficients, standard errors, and probabilities, as well as results for each diagnostic test. The diagnostic table includes a description of each test along with some guidelines for how to interpret test results.

The coefficient table includes computed coefficients, standard errors, and variable probabilities.

The diagnostic table includes notes for interpreting model diagnostic test results.

You may use the AICc value to compare regression models.

Additional resources

There are a number of good resources to help you learn more about OLS regression. Start by reading the Regression Analysis Basics documentation and/or watching the free one-hour ESRI Virtual Campus Regression Analysis Basics Web seminar. Next, work through a Regression Analysis tutorial.