Regression diagnostics
regstats(y,X,
model
)
stats = regstats(...)
stats = regstats(y,X,model
,whichstats
)
regstats(y,X,
performs
a multilinear regression of the responses in model
)y
on
the predictors in X
. X
is an n-by-p matrix
of p predictors at each of n observations. y
is
an n-by-1 vector of observed responses.
Note:
By default, |
The optional input model
controls
the regression model. By default, regstats
uses
a linear additive model with a constant term. model
can
be any one of the following:
'linear'
— Constant and
linear terms (the default)
'interaction'
— Constant,
linear, and interaction terms
'quadratic'
— Constant,
linear, interaction, and squared terms
'purequadratic'
— Constant,
linear, and squared terms
Alternatively, model
can be a matrix of model
terms accepted by the x2fx
function.
See x2fx
for a description of
this matrix and for a description of the order in which terms appear.
You can use this matrix to specify other models including ones without
a constant term.
With this syntax, the function displays a graphical user interface (GUI) with a list of diagnostic statistics, as shown in the following figure.
When you select check boxes corresponding to the statistics
you want to compute and click OK, regstats
returns
the selected statistics to the MATLAB^{®} workspace. The names of
the workspace variables are displayed on the right-hand side of the
interface. You can change the name of the workspace variable to any
valid MATLAB variable name.
stats = regstats(...)
creates the structure stats
,
whose fields contain all of the diagnostic statistics for the regression.
This syntax does not open the GUI. The fields of stats
are
listed in the following table.
Field | Description |
---|---|
Q | Q from the QR decomposition of the design matrix |
R | R from the QR decomposition of the design matrix |
beta | Regression coefficients |
covb | Covariance of regression coefficients |
yhat | Fitted values of the response data |
r | Residuals |
mse | Mean squared error |
rsquare | R^{2} statistic |
adjrsquare | Adjusted R^{2} statistic |
leverage | Leverage |
hatmat | Hat matrix |
s2_i | Delete-1 variance |
beta_i | Delete-1 coefficients |
standres | Standardized residuals |
studres | Studentized residuals |
dfbetas | Scaled change in regression coefficients |
dffit | Change in fitted values |
dffits | Scaled change in fitted values |
covratio | Change in covariance |
cookd | Cook's distance |
tstat | t statistics and p-values for coefficients |
fstat | F statistic and p-value |
dwstat | Durbin-Watson statistic and p-value |
Note that the fields names of stats
correspond
to the names of the variables returned to the MATLAB workspace
when you use the GUI. For example, stats.beta
corresponds
to the variable beta
that is returned when you
select Coefficients in the GUI and click OK.
stats = regstats(y,X,
returns
only the statistics that you specify in model
,whichstats
)whichstats
. whichstats
can
be a single character vector such as 'leverage'
or
a cell array of character vectors such as {'leverage' 'standres'
'studres'}
. Set whichstats
to 'all'
to
return all of the statistics.
Note: The F statistic is computed under the assumption that the model contains a constant term. It is not correct for models without a constant. The R^{2} statistic can be negative for models without a constant, which indicates that the model is not appropriate for the data. |
Open the regstats
GUI using data from hald.mat
:
load hald regstats(heat,ingredients,'linear');
Select Fitted Values and Residuals in the GUI:
Click OK to export the fitted values
and residuals to the MATLAB workspace in variables named yhat
and r
,
respectively.
You can create the same variables using the stats
output,
without opening the GUI:
whichstats = {'yhat','r'}; stats = regstats(heat,ingredients,'linear',whichstats); yhat = stats.yhat; r = stats.r;
[1] Belsley, D. A., E. Kuh, and R. E. Welsch. Regression Diagnostics. Hoboken, NJ: John Wiley & Sons, Inc., 1980.
[2] Chatterjee, S., and A. S. Hadi. "Influential Observations, High Leverage Points, and Outliers in Linear Regression." Statistical Science. Vol. 1, 1986, pp. 379–416.
[3] Cook, R. D., and S. Weisberg. Residuals and Influence in Regression. New York: Chapman & Hall/CRC Press, 1983.
[4] Goodall, C. R. "Computation Using the QR Decomposition." Handbook in Statistics. Vol. 9, Amsterdam: Elsevier/North-Holland, 1993.