RSTAP
Computes summary statistics for a polynomial regression model given the fit based on orthogonal polynomials.
Required Arguments
A — Vector of length NDEG containing constants used to generate orthogonal polynomials. (Input)
B — Vector of length NDEG containing constants used to generate orthogonal polynomials. (Input)
SMULTC — Multiplicative constant used to compute the scaled version of x, say z, on the interval ‑2 to 2, inclusive. (Input)
SADDC — Additive constant used to compute the scaled version of x(z) on the interval ‑2 to 2, inclusive. (Input)
SCOEF — Vector of length NDEG + 1 containing the regression coefficients of the fitted model using the scaled version of the original data. (Input)
SCOEF(1) is the estimated intercept. SCOEF(1 + i) contains the estimated coefficient for the i-th order orthogonal polynomial using z.
D — Vector of length NDEG + 1 containing the diagonal elements of the (diagonal) sums of squares and crossproducts matrix. (Input)
DFE — Degrees of freedom for error. (Input)
SSE — Sum of squares for error. (Input)
COEF — NDEG + 1 by 4 matrix containing statistics relating to the coefficients of the polynomial model. (Output)
Row 1 corresponds to the intercept. Row 1 + i corresponds to the coefficient of xi. The columns are described as follows:
|
Col. |
Description |
|
1 |
Estimated coefficient |
|
2 |
Estimated standard error of estimated coefficient |
|
3 |
t-statistic for the test that the coefficient is zero |
|
4 |
p‑value for the two-sided t test |
Optional Arguments
NDEG — Degree of the polynomial regression. (Input)
Default: NDEG = size (A,1).
LOF — Lack of fit test option. (Input)
Default: LOF = 0.
|
LOF |
Action |
|
0 |
No lack of fit test is performed. |
|
1 |
Lack of fit test is performed. |
DFPE — Degrees of freedom for pure error. (Input, if LOF = 1)
If LOF = 0, DFPE is not referenced.
Default: DFPE = 1.0.
SSPE — Sum of squares for pure error. (Input, if LOF = 1)
If LOF = 0, SSPE is not referenced.
Default: SSPE = 0.0.
IPRINT — Printing option. (Input)
Default: IPRINT = 0.
|
IPRINT |
Action |
|
0 |
No printing is performed. |
|
1 |
AOV, SQSS, COEF are printed. |
AOV — Vector of length 15 that contains statistics relating to the analysis of variance. (Output)
|
I |
AOV(I) |
|
1 |
Degrees of freedom for the model |
|
2 |
Degrees of freedom for error |
|
3 |
Total (corrected) degrees of freedom |
|
4 |
Sum of squares for the model |
|
5 |
Sum of squares for error |
|
6 |
Total (corrected) sum of squares |
|
7 |
Model mean square |
|
8 |
Error mean square |
|
9 |
Overall F -statistic |
|
10 |
p‑value |
|
11 |
R2 (in percent) |
|
12 |
Adjusted R2 (in percent) |
|
13 |
Estimate of the standard deviation |
|
14 |
Overall mean of y |
|
15 |
Coefficient of variation (in percent) |
SQSS — NDEG by 4 matrix containing sequential statistics for the polynomial model. (Output)
Row i corresponds to xi(i = 1, 2, …, NDEG). The columns are described as follows:
|
Col. |
Description |
|
1 |
Degrees of freedom |
|
2 |
Sum of squares |
|
3 |
F -statistic |
|
4 |
p‑value |
LDSQSS — Leading dimension of SQSS exactly as specified in the dimension statement of the calling program. (Input)
Default: LDSQSS = size (SQSS,1).
LDCOEF — Leading dimension of COEF exactly as specified in the dimension statement of the calling program. (Input)
Default: LDCOEF = size (COEF,1).
TLOF — NDEG by 4 matrix containing tests of lack of fit for each degree of the polynomial. (Output, if LOF = 1)
If LOF = 0, TLOF is not referenced and can be a 1 by 1 array. Row i corresponds to xi(i = 1, 2, …, NDEG). The columns are described as follows:
|
Col. |
Description |
|
1 |
Degrees of freedom |
|
2 |
Lack of fit sum of squares |
|
3 |
F test for lack of fit of the polynomial model of degree i |
|
4 |
p‑value for the F test |
LDTLOF — Leading dimension of TLOF exactly as specified in the dimension statement of the calling program. (Input)
Default: LDTLOF = size (TLOF,1).
FORTRAN 90 Interface
Generic: CALL RSTAP (A, B, SMULTC, SADDC, SCOEF, D, DFE, SSE, COEF [, …])
Specific: The specific interface names are S_RSTAP and D_RSTAP.
FORTRAN 77 Interface
Single: CALL RSTAP (NDEG, A, B, SMULTC, SADDC, SCOEF, D, DFE, SSE, LOF, DFPE, SSPE, IPRINT, AOV, SQSS, LDSQSS, COEF, LDCOEF, TLOF, LDTLOF)
Double: The double precision name is DRSTAP.
Description
Routine RSTAP transforms a polynomial regression model, fitted using orthogonal polynomials, into a polynomial function of the original independent variable. In addition, summary statistics (analysis of variance, t tests, tests for lack of fit) are computed. Results from routine RFORP, which produces the fit using orthogonal polynomials, are used for input.
The fitted model from RFORP is
where the zi’s are the settings of the independent variable x scaled to the interval [‑2, 2] and where the pj(z)’s are the orthogonal polynomials. The “XT X” matrix for this model is a diagonal matrix with elements dj (stored in D). The orthogonal polynomials can be expressed as
First, RSTAP computes
to produce the fit for the polynomial function in terms of the scaled independent variable as given by
The variances and covariances for the estimated coefficients in this model are given by
Second, RSTAP computes
as a linear combination of the
by the formula
in order to produce the fit for the polynomial function in terms of the original independent variable as given by
The variance of
computed from the variances and covariances of the
using the usual formula for computing variances of linear combinations of correlated random variables. The sequential sum of squares due to xj(stored in SQSS) is computed by
Comments
Workspace may be explicitly provided, if desired, by use of R2TAP/DR2TAP.
The reference is:
CALL R2TAP (NDEG, A, B, SMULTC, SADDC, SCOEF, D, DFE, SSE, LOF, DFPE, SSPE, IPRINT, AOV, SQSS, LDSQSS, COEF, LDCOEF, TLOF, LDTLOF, WK)
The additional argument is:
WK — Work vector of length (NDEG + 1) * (NDEG + 7).
Example
A polynomial model is fitted to data discussed by Neter and Wasserman (1974, pages 279‑285). The data set contains the response variable y measuring coffee sales (in hundred gallons) and the number of self-service coffee dispensers. Responses for fourteen similar cafeterias are in the data set and some of the cafeterias have the same number of dispensers so that lack of fit of the model can be assessed.
USE RSTAP_INT
USE RFORP_INT
IMPLICIT NONE
INTEGER LDCOEF, LDSQSS, LDTLOF, LDX, MAXDEG, NCOL, NOBS, J
PARAMETER (MAXDEG=2, NCOL=2, NOBS=14, LDCOEF=MAXDEG+1, &
LDSQSS=MAXDEG, LDTLOF=MAXDEG, LDX=NOBS)
!
INTEGER IND, IPRINT, IRSP, LOF, NDEG, NRMISS
REAL A(MAXDEG), AOV(15), B(MAXDEG), COEF(MAXDEG+1,4), &
CRIT, D(MAXDEG+1), DFE, DFPE, SADDC, SCOEF(MAXDEG+1), &
SMULTC, SQSS(LDSQSS,4), SSE, SSPE, TLOF(MAXDEG,4), &
X(LDX,NCOL)
!
DATA (X(1,J),J=1,2) /0.0, 508.1/
DATA (X(2,J),J=1,2) /5.0, 787.6/
DATA (X(3,J),J=1,2) /0.0, 498.4/
DATA (X(4,J),J=1,2) /1.0, 568.2/
DATA (X(5,J),J=1,2) /2.0, 651.7/
DATA (X(6,J),J=1,2) /7.0, 854.7/
DATA (X(7,J),J=1,2) /2.0, 657.0/
DATA (X(8,J),J=1,2) /4.0, 755.3/
DATA (X(9,J),J=1,2) /6.0, 831.8/
DATA (X(10,J),J=1,2) /4.0, 758.9/
DATA (X(11,J),J=1,2) /5.0, 792.1/
DATA (X(12,J),J=1,2) /6.0, 841.4/
DATA (X(13,J),J=1,2) /7.0, 871.4/
DATA (X(14,J),J=1,2) /1.0, 577.3/
!
IRSP = 2
IND = 1
LOF = 1
CALL RFORP (X, IRSP, IND, MAXDEG, NDEG, A, B, SCOEF, D, LOF=LOF, &
SMULTC=SMULTC, SADDC=SADDC, DFE=DFE, SSE=SSE, &
DFPE=DFPE, SSPE=SSPE)
!
IPRINT = 1
CALL RSTAP (A, B, SMULTC, SADDC, SCOEF, D, DFE, SSE, COEF, &
NDEG=NDEG, LOF=LOF, DFPE=DFPE, SSPE=SSPE, IPRINT=IPRINT, &
AOV=AOV, SQSS=SQSS, TLOF=TLOF)
END
Output
R-squared Adjusted Est. Std. Dev. Coefficient of
(percent) R-squared of Model Error Mean Var. (percent)
99.685 99.628 8.037 711.0 1.13
* * * Analysis of Variance * * *
Sum of Mean Prob. of
Source DF Squares Square Overall F Larger F
Regression 2 225031.9 112515.9 1741.748 0.0000
Residual 11 710.6 64.6
Corrected Total 13 225742.5
* * * Inference on Coefficients * * *
Standard Prob. of
Coef. Estimate Error t-statistic Larger |t|
1 503.3 4.791 105.054 0.0000
2 78.9 3.453 22.865 0.0000
3 -4.0 0.482 -8.242 0.0000
* * * Sequential Statistics * * *
Degree of Degrees of Sum of Prob. of
Polynomial Freedom Squares F-statistic Larger F
1 1 220644.1 3415.574 0.0000
2 1 4387.7 67.922 0.0000
* * * Tests of Lack of Fit * * *
Degree of Degrees of Sum of Prob. of
Polynomial Freedom Squares F-statistic Larger F
1 5 4793.7 22.031 0.0004
2 4 406.0 2.332 0.1547
