This page shows an example simple regression analysis with footnotes explaining the output. The analysis uses a data file about scores obtained by elementary schools, predicting api00 from enroll using the following SPSS commands.
regression /dependent api00 /method=enter enroll.
The output of this command is shown below, followed by explanations of the output.
|Model||Variables Entered||Variables Removed||Method|
|a All requested variables entered.|
|b Dependent Variable: API00|
|Model||Rb||R Squarec||Adjusted R Squared||Std. Error of the Estimatee|
|a Predictors: (Constant), ENROLL|
|Modelf||Sum of Squaresg||dfh||Mean Squarei||Fj||Sig.j|
|a Predictors: (Constant), ENROLL|
|b Dependent Variable: API00|
|Unstandardized Coefficients||Standardized Coefficients||to||Sig.o|
|a Dependent Variable: API00|
a. This is a summary of the analysis, showing that api00 was the dependent variable and enroll was the predictor variable.
b. R is the square root of R Square (shown in the next column).
c. R Square is the proportion of variance in the dependent variable (api00) which can be predicted from the independent variable (enroll). This value indicates that 10% of the variance in api00 can be predicted from the variable enroll.
d. Adjusted R square. As predictors are added to the model, each predictor will explain some of the variance in the dependent variable simply due to chance. One could continue to add predictors to the model which would continue to improve the ability of the predictors to explain the dependent variable, although some of this increase in R-square would be simply due to chance variation in that particular sample. The adjusted R-square attempts to yield a more honest value to estimate the R-squared for the population. The value of R-square was .10, while the value of Adjusted R-square was .099. Adjusted R-squared is computed using the formula 1 – ( (1-R-sq)(N-1 / N – k – 1) ). From this formula, you can see that when the number of observations is small and the number of predictors is large, there will be a much greater difference between R-square and adjusted R-square (because the ratio of (N-1 / N – k – 1) will be much less than 1. By contrast, when the number of observations is very large compared to the number of predictors, the value of R-square and adjusted R-square will be much closer because the ratio of (N-1)/(N-k-1) will approach 1.
e. Std. Error of the Estimate is the standard deviation of the error term, and is the square root of the Mean Square Residual (or Error)
f. This shows the model number (in this case we ran only one model, so it is model #1). Also, this column shows the source of variance, Regression, Residual, and Total. The Total variance is partitioned into the variance which can be explained by the indendent variables (Regression) and the variance which is not explained by the independent variables (Residual). Note that the Sums of Squares for the Regression and Residual add up to the Total Variance, reflecting the fact that the Total Variance is partitioned into Regression and Residual variance.
g. These are the Sum of
Squares associated with the three sources of variance, Total, Regression & Residual.
These can be computed in many ways. Conceptually, these formulas
can be expressed as:
SSTotal. The total variability around the mean. S(Y – Ybar)2.
SSResidual. The sum of squared errors in prediction. S(Y – Ypredicted)2.
SSRegression. The improvement in prediction by using the predicted value of Y over just using the mean of Y. Hence, this would be the squared differences between the predicted value of Y and the mean of Y, S(Ypredicted – Ybar)2. Another way to think of this is the SSRegression is SSTotal – SSResidual. Note that the SSTotal = SSRegression + SSResidual. Note that SSRegression / SSTotal is equal to .10, the value of R-Square. This is because R-Square is the proportion of the variance explained by the independent variables, hence can be computed by SSRegression / SSTotal.
h. These are the degrees of freedom associated with the sources of variance. The total variance has N-1 degrees of freedom. In this case, there were N=400 observations, so the DF for total is 399. The regression degrees of freedom corresponds to the number of predictors minus 1 (K-1). You may think this would be 1-1 (since there was 1 independent variable in the model statement, enroll). But, the intercept is automatically included in the model (unless you explicitly omit the intercept). Including the intercept, there are 2 predictors, so the model has 2-1=1 degree of freedom. The Residual degrees of freedom is the DF total minus the DF model, 399 – 1 is 398.
i. These are the Mean Squares, the Sum of Squares divided by their respective DF. For the Regression, 817326.293 / 1 is equal to 817326.293. For the Residual, 7256345.7 / 398 equals 18232.0244. These are computed so you can compute the F ratio, dividing the Mean Square Model by the Mean Square Residual to test the significance of the predictor(s) in the model.
j. The F Value is the Mean Square Model (817326.293) divided by the Mean Square Residual (18232.0244), yielding F=44.83. The p value associated with this F value is very small (0.0000). These values are used to answer the question "Do the independent variables reliably predict the dependent variable?". The p value is compared to your alpha level (typically 0.05) and, if smaller, you can conclude "Yes, the independent variables reliably predict the dependent variable". You could say that the variable enroll can be used to reliably predict api00 (the dependent variable). If the p value were greater than 0.05, you would say that the independent variable does not show a significant relationship with the dependent variable, or that the independent variable does not reliably predict the dependent variable.
k. This column shows the predictor variables below it (enroll). The last variable (_cons) represents the constant, also referred to in textbooks as the Y intercept, the height of the regression line when it crosses the Y axis.
l. These are the values for the regression equation for predicting the dependent variable from the independent variable. The regression equation is presented in many different ways, for example…
Ypredicted = b0 + b1*x1
The column of estimates (coefficients or parameter estimates, from here on labeled coefficients) provides the values for b0 and b1 for this equation. Expressed in terms of the variables used in this example, the regression equation is
api00Predicted = 744.25 – .20*enroll
Thise estimate tells you about the relationship
between the independent variable and the dependent variable. This estimate indicates
the amount of increase in api00 that would be predicted by a 1 unit increase in the
predictor. Note: If an independent variable is not significant, the
coefficient is not significantly different from 0, which should be taken into account
when interpreting the coefficient. (See the columns with the t value and p value
about testing whether the coefficients are significant).
enroll – The coefficient (parameter estimate) is -.20. So, for every unit increase in enroll, a -.20 unit decrease in api00 is predicted.
m. These are the standard errors associated with the coefficients. The standard error is used for testing whether the parameter is significantly different from 0 by dividing the parameter estimate by the standard error to obtain a t value (see the column with t values and p values). The standard errors can also be used to form a confidence interval for the parameter, as shown in the last 2 columns of this table.
n. These are the Standardized regression coefficients. These are the coefficients that you would obtain if the predictors and the outcome variables were standardized prior to the analysis. Since all of the predictors are standardized, they are measured in the same units, so the standardized regression coefficients are useful for comparing the size of the coefficients across variables. Since the variables are measured in standard units, a one unit change corresponds to a one standard deviation change. For the variable enroll, we would interpret the coefficient as saying "for a one standard deviation increase in enroll, we would expect a -.318 standard deviation decrease in api00.
o. These columns
provide the t value and 2 tailed p value used in testing the null hypothesis that the
coefficient/parameter is 0. If you use a 2 tailed test, then you would compare each
p value to your preselected value of alpha. Coefficients having p values less than
alpha are significant. For example, if you chose alpha to be 0.05, coefficients
having a p value of 0.05 or less would be statistically significant (i.e. you can reject
the null hypothesis and say that the coefficient is significantly different from 0).
If you use a 1 tailed test (i.e., you predict that the parameter will go in a
particular direction), then you can divide the p value by 2 before comparing it to your
preselected alpha level. With a 2 tailed test and alpha of 0.05, you can reject the
null hypothesis that the coefficient for enroll is equal to 0. The coefficient of
-.20 is significantly different from 0. Using a 2 tailed test and alpha of
0.01, the p value of 0.000 is smaller than 0.01 and the coefficient for enroll would still
be significant at the 0.01 level.
The constant is significantly different from 0 at the 0.05 alpha level. However, having a significant intercept is seldom interesting.