RINCF

Performs response control given a fitted simple linear regression model.

Required Arguments

SUMWTF — Sum of products of weights with frequencies from the fitted regression. (Input, if INTCEP = 1)
In the ordinary case when weights and frequencies are all one, SUMWTF equals the number of observations.

DFE — Degrees of freedom for error from the fitted regression. (Input)

B — Vector of length INTCEP + 1 containing a least-squares solution for the intercept and slope. (Input)

 

INTCEP

Intercept

Slope

0

 

B(1)

1

B(1)

B(2)

XYMEAN — Vector of length 2 containing the variable means. (Input)
XYMEAN(1) is the independent variable mean. XYMEAN(2) is the dependent variable mean. If INTCEP = 0, XYMEAN is not referenced and can be a vector of length one.

SSX — Sum of squares for the independent variable. (Input)
If INTCEP = 1, SSX is the sums of squares of deviations of the independent variable from its mean. Otherwise, SSX is not corrected for the mean.

S2 — s2, the estimate of σ2 from the fitted regression. (Input)

YLOWER — Lower limit for the response. (Input)

YUPPER — Upper limit for the response. (Input)

XLOWER — Lower limit on the independent variable for controlling the response. (Output)

XUPPER — Upper limit on the independent variable for controlling the response. (Output)

Optional Arguments

INTCEP — Intercept option. (Input)
Default: INTCEP = 1.

 

INTCEP

Action

0

An intercept is not in the model.

1

An intercept is in the model.

SWTFY0 — S2/SWTFY0 is the estimated variance of the future response (or future response mean) that is to be controlled. (Input)
In the ordinary case, when weights and frequencies are all one, SWTFY0 is the number of observations in the response mean that is to be controlled. SWTFY0 = 0.0 means the true response mean is to be controlled.
Default: SWTFY0 = 0.0.

CONPER — Confidence level for a two-sided response control, in percent. (Input)
CONPER percent limits are computed; hence, CONPER must be greater than or equal to 0.0 and less than 100.0. CONPER often will be 90.0, 95.0, or 99.0. For one-sided control with confidence level ONECL, where ONECL is greater than or equal to 50.0 and less than 100.0, set CONPCM = 100.0 ‑ 2.0 * (100.0 ‑ ONECL).
Default: CONPER = 95.0.

FORTRAN 90 Interface

Generic: CALL RINCF (SUMWTF, DFE, B, XYMEAN, SSX, S2, YLOWER, YUPPER, XLOWER, XUPPER [, …])

Specific: The specific interface names are S_RINCF and D_RINCF.

FORTRAN 77 Interface

Single: CALL RINCF (SUMWTF, DFE, INTCEP, B, XYMEAN, SSX, S2, SWTFY0, CONPER, YLOWER, YUPPER, XLOWER, XUPPER)

Double: The double precision name is DRINCF.

Description

Routine RINCF estimates settings of the independent variable that restrict, at a specified confidence percentage, the average of k randomly drawn responses to a given acceptable range (or the true mean response to a given acceptable range), using a fitted simple linear regression model. The results of routine RLINE or RONE can be used for input into RINCF. The simple linear regression model is assumed:

yi= β0 + β1xi+ ɛ ii = 1, 2, …, n + k

where the ɛi’s are independently distributed normal errors with mean zero and variance σ2/wi. Here, n is the total number of observations used in the fit of the line, i.e., n = DFE + INTCEP + 1. Also, k is the number of additional responses whose average is to be restricted to the specified range. The wi’s are the weights.

The methodology is based on Graybill (1976, pages 280‑283). The estimate of σ2, s2 (stored in S2), is the usual estimate of σ2 from the fitted regression based on the first n observations. First, a test of the hypothesis H0 : β1 = 0 vs. Ha : β1 ≠ 0 at level α = 1 ‑ CONPER/100 is performed. If H0 is accepted, the model becomes yi = β0 + ɛi, and limits for x to control the response are meaningless since x is no longer in the model. In this case, a type 4 fatal error is issued. If H0 is rejected and is positive, a lower limit (upper limit) for x stored in XLOWER(XUPPER) is computed for the case where SWTFY0 is positive by

 

where y0 is the value stored in YLOWER(YUPPER) and where

 

and t is the 50 + CONPER/2 percentile of the t distribution with DFE degrees of freedom. In the formula, the symbol ± is used to indicate that + is used to compute XLOWER with y0 = YLOWER, and ‑ is used to compute XUPPER with y0 = YUPPER. If H0 is rejected and is negative, a lower limit (upper limit) for x stored in XLOWER(XUPPER) is computed for the case where SWTFY0 is positive by a small modification. In particular, the symbol ± is then taken so that + is used to compute XLOWER with y0 = YUPPER, and ‑ is used to compute XUPPER with y0 = YLOWER. These limits actually have a confidence coefficient less than that specified by CONPER.

In the weighted case, which was discussed earlier, the means (stored in XYMEAN) and the sum of squares for x (stored in SSX) are all weighted. When the variances of the ɛi’s are all equal, ordinary least squares must be used, this corresponds to all wi = 1.

The previous discussion can be generalized to the case where an intercept is not in the model. The necessary modifications are to let and to replace the first term under the square root symbol by zero, by zero, and by zero.

In order to restrict the true mean response to a specified range, i.e, when SWTFY0 is zero, the formulas are modified by replacing the second term under the square root symbol with zero.

Comments

Informational errors

 

Type

Code

Description

4

1

The slope is not significant at the (100 ‑ CONPER) percent level. Control limits cannot be obtained.

4

2

The computed lower limit, XLOWER, exceeds the computed upper limit, XUPPER. No satisfactory settings of the independent variable exist to control the response as specified.

Example

This example estimates the settings of the independent variable that restrict, at 97.5% confidence, the true mean response to a upper bound of ‑4.623, using a fitted simple linear regression model. The fitted model excludes the intercept term. To accomplish one-sided control, CONPER is set to 100 ‑ 2(100 ‑ 97.5) = 95, and YLOWER is set to an arbitrary value less than YUPPER. The output for XLOWER furnishes the lower bound for x necessary to control y.

 

USE RINCF_INT

USE UMACH_INT

 

IMPLICIT NONE

INTEGER INTCEP

PARAMETER (INTCEP=0)

!

INTEGER NOUT

REAL B(INTCEP+1), CONPER, DFE, ONECL, S2, SSX, SUMWTF, &

SWTFY0, XLOWER, XUPPER, XYMEAN(1), YLOWER, YUPPER

!

DATA B/-.079829/

!

SUMWTF = 25.0

DFE = 24.0

SSX = 76323.0

S2 = 0.7926

SWTFY0 = 0.0

ONECL = 97.5

CONPER = 100.0 - 2*(100.0-ONECL)

YUPPER = -4.623

YLOWER = -9.0

CALL RINCF (SUMWTF, DFE, B, XYMEAN, SSX, S2, YLOWER, YUPPER, &

XLOWER, XUPPER, INTCEP=INTCEP, CONPER=CONPER)

CALL UMACH (2, NOUT)

WRITE (NOUT,*) 'XLOWER = ', XLOWER, ' XUPPER = ', XUPPER

END

Output

 

XLOWER = 63.1747 XUPPER = 104.07