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University of North Carolina Chapel Hill
Soci708-001 Statistics for Sociologists
Fall 2009
Professor François Nielsen
Stata Commands for Module 10 – Inference for Regression
For further information on any command in this handout, simply type help
followed by the name of the command in Stata.
See also the Stata and SAS Guide pdf (click on Documents in side bar; guide
is linked under Software Documentation).
1 Statistical Functions in Stata
The following statistical functions in Stata are useful for regression work. The
regression printout itself usually comprises all necessary statistics.
1.1 Normal Distribution Functions
The function normal(z) returns P ( Z ≤ z ), the area under the standard normal
curve to the left of z. (Compare with Table A.)
. display normal(1.207) .
The function invnormal(p) returns z such that P ( Z ≤ z ) = p , i.e. such that
the area under the standard normal curve to the left of z is p. (Compare with
Table A and Table D (bottom row).)
. display invnormal(0.975)
1.2 Student t Distribution Functions
The function ttail(df, t) returns P ( T > t ), the area under the Student’s t
distribution with df degrees of freedom to the right of t. (Compare with Table
D.).
. display ttail(7, 1.960) .
The function invttail(df, p) returns t such that P ( T > t ) = p , i.e. t
such that the area under Student’s t distribution with df degrees of freedom to
the right of t is p. (Compare with Table D.)
. display invttail(7, 0.025)
1.3 F Distribution Functions
The function Ftail(n1, n2, f) returns P ( F > f ), the area under the F
distribution with n1 and n2 degrees of freedom to the right of f. (Compare
with Table E.)
. display Ftail(1, 14, 21.55) .
The function invFtail(n1, n2, p) returns f such that P ( F > f ) = p ,
i.e. f such that the area under the F distribution with n1 and n2 degrees of
freedom to the right of f is p. (Compare with Table D.)
. display invFtail(1, 14, .00038068)
2 Simple Regression With Direct Data Input
Simple example where x is years of education and y is yearly income in dollars.
. input x
x
- 12
- 13
- 12
- 14
- 12
- 15
- 12
- end . input y
y
- 20000
- 22000
- 23000
- 25000
- 18000
- 30000
- 26000
- end . reg y x
Source | SS df MS Number of obs = 7 -------------+------------------------------ F( 1, 5) = 5. Model | 51843318 1 51843318 Prob > F = 0. Residual | 43870967.7 5 8774193.55 R-squared = 0. -------------+------------------------------ Adj R-squared = 0. Total | 95714285.7 6 15952381 Root MSE = 2962.
y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- x | 2419.355 995.3064 2.43 0.059 -139.1616 4977.
Figure 1: Plot of fat gain by nonexercise activity with lowess curve.
4 Graphs and Residual Analysis for Simple Regression
We continue with the previous example to show some useful commands.
To check linearity of the regression we may want to do a lowess plot of the
response variable fat against the explanatory variable nea. (Lowess is a type
of nonparametric regression used to show the main trend in the data.)
. lowess fat nea
The lowess plot is shown in Figure 1. The linear appearance of the lowess
curve gives us confidence that the relationship between fat and nea is linear.
To make a scatterplot with the regression line, use the following command.
When the graph appear you can save it in a variety of formats from the graph
window. This produces the graph shown in Figure 2.
. twoway (scatter fat nea) (lfit fat nea)
Now make a similar plot using the residuals of the regression of fat on nea
on the y-axis. First we repeat the regression command (which is not necessary
if you have already run it). Then we get to the residuals by creating a variable
fatresid (my invented name) and telling Stata that this variable corresponds
to the residuals of the regression we just ran, with the command predict
fatresid, residuals. Then we create another variable called fatpredict
(which will be essentially equivalent to the explanatory variable nea), using
the command predict fatpredict, xb. Finally we do a plot of fatresid
(the residual e ) against fatpredict (the predictor ˆ y ). The plot is shown in
Figure 3.
. reg fat nea . predict fatresid, residuals
Figure 2: Scatterplot of fat gain by nonexercise activity with fitted regression
line.
. predict fatpredict, xb . twoway (scatter fatresid fatpredict), yline(0)
We further analysze the distribution of residuals by creating a histogram
(with superposed kernel density) (Figure 4) and a normal quantile plot (Fig-
ure 5). We note that except for one rather large positive residual the residuals
in this case appear fairly normal.
. histogram fatresid, kdensity (bin=4, start=-1.1090604, width=.68825009) . qnorm fatresid
Finally another useful plot is the linear prediction plot with confidence lim-
its for the mean response. This is obtained with the following command. Make
sure you place the lfitci fat nea part first, and the scatter fat nea
part second, as otherwise the confidence band produced by lfitci will hide
the data points. The plot is shown in Figure 6.
. twoway (lfitci fat nea) (scatter fat nea)
Finally the last command shows the prediction plot with both the 95% con-
fidence limits for the mean response (lfitci with default stdp option; com-
pare with IPS6e Figure 10.9 p.573) and the the 95% confidence limits for in-
dividual responses (lfitci, stdf; compare with IPS6e Figure 10.10 p.575).
The latter confidence includes uncertainty due to prediction of the mean and
uncertainty due to prediction of individual responses and is useful to detect
outliers. The plot is shown in Figure 7.
. twoway (lfitci fat nea) (lfitci fat nea, stdf ciplot(rline)) (scatter fat nea)
Figure 5: Normal quantile plot of fat gain residuals fatresid.
Figure 6: Linear predictor plot of fat with nea with 95% confidence limits for
the mean response.
Figure 7: Linear predictor plot of fat with nea with 95% confidence limits
for the mean response (lfitci with default stdp option; shaded area) and
the 95% confidence limits for individual responses (lfitci, stdf; broader
confidence band).
