Performs response control given a fitted simple linear regression model.
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)
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.
Generic: CALL RINCF (SUMWTF, DFE, B, XYMEAN, SSX, S2, YLOWER, YUPPER, XLOWER, XUPPER [,…])
Specific: The specific interface names are S_RINCF and D_RINCF.
Single: CALL RINCF (SUMWTF, DFE, INTCEP, B, XYMEAN, SSX, S2, SWTFY0, CONPER, YLOWER, YUPPER, XLOWER, XUPPER)
Double: The double precision name is DRINCF.
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 i i = 1, 2, …, n + k
where the ɛ i’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 + ɛ
i, 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 ɛ i’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.
Informational errors
Type Code
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.
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
XLOWER = 63.1747 XUPPER = 104.07
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