The Residual degrees of
freedom is the DF total minus the DF model, 199 - 4 is 195.
d. MS - These are the Mean Squares, the Sum of Squares
divided by their respective DF. For the Model,
9543.72074 / 4 = 2385.93019. For the Residual,
9963.77926 / 195 = 51.0963039. 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 predictors
in the model.
Overall Model Fit
Number of obse = 200
F( 4, 195)f = 46.69
Prob > Ff = 0.0000
R-squaredg = 0.4892
Adj R-squaredh = 0.4788
Root MSEi = 7.1482
e. Number of obs - This is the number of observations used
in the regression
analysis.
f. F and Prob > F - The F-value is
the Mean Square Model (2385.93019) divided by the Mean Square
Residual (51.0963039), yielding F=46.69. 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 group of
variables math and female can be used to reliably
predict science (the dependent variable).
If the p-value were greater than 0.05, you would say that the group
of independent variables does not show a statistically significant
relationship with the dependent variable, or that the group of
independent variables does not reliably predict the dependent
variable. Note that this is an overall
significance test assessing whether the group of independent
variables when used together reliably predict the dependent
variable, and does not address the ability of any of the particular
independent variables to predict the dependent
variable. The ability of each individual
independent variable to predict the dependent variable is addressed
in the table below
where each of the individual variables are listed.
g. R-squared - R-Squared is the proportion of variance
in the dependent
variable (science) which can be predicted from the
independent variables (math, female, socst and
read). This value indicates that 48.92% of
the variance in science scores
can be predicted from the variables math, female,
socst and read. Note that this is
an overall measure of the strength of association, and does not
reflect the extent to which any particular independent variable is
associated with the dependent variable.
h. Adj R-squared - 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 .4892, while the value of Adjusted R-square was
.4788 Adjusted R-squared is computed using the
formula 1 - ((1 - Rsq)((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
greater 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.
i. Root MSE - Root
MSE is the
standard deviation of the error term, and is the square root of the
Mean Square Residual (or Error).
Parameter Estimates
------------------------------------------------------------------------------
sciencej | Coef.k Std. Err.l tm P>|t|m [95% Conf. Interval]n
-------------+----------------------------------------------------------------
math | .3893102 .0741243 5.25 0.000 .243122 .5354983
female | -2.009765 1.022717 -1.97 0.051 -4.026772 .0072428
socst | .0498443 .062232 0.80 0.424 -.0728899 .1725784
read | .3352998 .0727788 4.61 0.000 .1917651 .4788345
_cons | 12.32529 3.193557 3.86 0.000 6.026943 18.62364
------------------------------------------------------------------------------
j. science - This column shows the dependent variable at
the top (science) with the predictor variables below it
(math, female, socst, read and
_cons). 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. In other words,
this is the predicted value of science when all other
variables are 0.
k. Coef. - 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 + b2*x2 + b3*x3 + b4*x4
The column of estimates (coefficients or parameter estimates,
from here on labeled coefficients) provides the values for b0, b1,
b2, b3 and b4 for this equation. Expressed
in terms of the
variables used in this example,
the regression equation is
sciencePredicted = 12.32529 + .3893102*math +
-2.009765*female+.0498443*socst+.3352998*read
These estimates tell you about the relationship between the
independent variables and the dependent variable.
These estimates tell the amount of increase in science scores
that would be predicted by a 1 unit increase in the
predictor. Note: For the independent variables
which are not significant, the coefficients are not significantly
different from 0, which should be taken into account when
interpreting the coefficients. (See the columns
with the t-value and p-value about testing whether the coefficients
are significant).
math
- The coefficient (parameter estimate) is
.3893102. So, for every unit (i.e., point, since
this is the metric in which the tests
are measured) increase in math, a
.3893102 unit increase in science
is predicted, holding all other variables constant. (It does not
matter at what value you hold the other variables constant, because
it is a linear model.) Or, for every increase of
one point on the math test, your science score is predicted
to be higher by .3893102 points. This is
significantly different from 0.
female - For every unit
increase in female,
there is a -2.009765 unit decrease in the predicted
science score, holding all other variables
constant. Since female is coded 0/1
(0=male, 1=female) the interpretation can be put more
simply. For females the predicted science score
would be 2 points lower than for males. The
variable female is technically not statistically
significantly different from 0, because the p-value is greater than
.05. However, .051 is so close to .05 that some
researchers would still consider it to be statistically
significant.
socst - The coefficient
for socst is .0498443. This means that for
a 1-unit increase in the social
studies score, we expect an approximately .05 point increase
in the science
score. This is not statistically significant;
in other words,
.0498443 is not different from 0.
read - The coefficient
for read is .3352998. Hence, for every
unit increase in reading score we
expect a .34 point increase in the science
score. This is statistically significant.
l. Std. Err. - 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 two columns of this table.
m. t and P>|t| - 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 statistically 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 may reject the null hypothesis
that the coefficient for female is equal to
0. The coefficient of -2.009765 is significantly
greater than 0. However, if you used a 2-tailed
test and alpha of 0.01, the p-value of .0255 is greater than 0.01
and the coefficient for female would not be significant at
the 0.01 level. Had you predicted that this coefficient would be
positive (i.e., a one tail test), you would be able to divide the
p-value by 2 before comparing it to alpha. This
would yield a 1-tailed p-value of 0.00945, which is less than 0.01
and then you could conclude that this coefficient is greater than 0
with a one tailed alpha of 0.01.
The coefficient for
math is significantly different from 0 using alpha of 0.05
because its p-value is 0.000, which is smaller than 0.05.
The coefficient for
socst (.0498443) is not statistically significantly
different from 0 because its p-value is definitely larger than
0.05.
The coefficient for
read (.3352998) is
statistically significant because
its p-value of 0.000 is less than .05.
The constant (_cons) is
significantly different from 0 at the 0.05 alpha level. However,
having a significant intercept is seldom interesting.
n. [95% Conf. Interval] - This shows a 95% confidence
interval for the coefficient. This is very useful
as it helps you understand how high and how low the actual
population value of the parameter might be. The
confidence intervals are related to the p-values such that the
coefficient will not be statistically significant if the confidence
interval includes 0. If you look at the
confidence interval for female, you will see that it just
includes 0 (-4 to .007). Because .007 is so close
to 0, the p-value is close to .05. If the upper
confidence level had been a little smaller, such that it did not
include 0, the coefficient for female would have been
statistically significant. Also, consider the
coefficients for female (-2) and read
(.34). Immediately you see that the estimate for
female is so much bigger, but examine the confidence
interval for it (-4 to .007). Now examine the
confidence interval for read (.19 to .48).
Even though female has a bigger coefficient (in absolute
terms) it could be as small as -4. By contrast, the lower
confidence level for read is .19, which is still above
0. So, even though female has a bigger
coefficient, read is significant and even the smallest value
in the confidence
interval is still higher than 0. The same cannot
be said about the coefficient for socst.
Such confidence intervals help you to put the estimate from the
coefficient into perspective by seeing how much the value could
vary.
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