national multiple family submetering and allocation billing program ...

particular property (e.g., the monthly rent of units in an individually-owned complex). The
cumulative effect of these missing data often resulted in a dramatically lower number of
properties being included in the analysis than the number of properties for which water use data
were available. Thus, the models calculated through stepwise regression only used the cases
where all the variables to be examined were present for that case, even if a variable was
ultimately eliminated from inclusion in the model. As both a test of the appropriateness of the
model, and to check for any other variable that sometimes can be significantly associated with a
dependent variable even if an automated method such as stepwise regression does not detect it,
many regression models were examined using a method that required entry of certain variables to
choose the most predictive models presented in Chapter 6.
The statistics produced for regression equations include a test of the hypothesis that there
is no relationship between the dependent variable and the predictor variables. The results of this
test are reported as an F-statistic with an associated p-value. In general, only models with a p-
value of 0.05 or less are considered “significant,” meaning that if there were no difference, the
probability of seeing a result as or more extreme than that seen in the sample was less than 5%.
In addition, an adjusted R-squared is calculated, which can be interpreted as the proportion of the
variability in the dependent variable accounted for by the factors included in the regression
model.
Regression coefficients are calculated for each predictor variable in the model. These
coefficients can be interpreted as a “slope,” that is, for every unit change in the predictor
variable, the independent variable would change by the amount of the regression coefficient. A
test of statistical significance is calculated for each regression coefficient, with a corresponding
p-value.
The fit of the model and the appropriateness of the variables for inclusion in the model
can be tested by examining a scatter plot of the predicted values (usually on the x-axis) and the
residual values, usually on the y-axis. A predicted value for the dependent variable can be
calculated for each case, given the values the independent variables in the model for each case.
The residual values are the difference between the actual value of the dependent variable for a
case and the predicted value. In a perfect model the residual value would be zero and all points
would lie on the x-axis. If there is not an abnormal distribution of the dependent variable or of
the other variables included in the regression model, the scatter plot will resemble a “cloud” or a
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