Computes statistics related to a regression fit given the coefficient estimates
and the R matrix.
IRBEF Index
vector of length |IEF|
+ 1. (Input, if IEF
is positive.)
For i = 1, 2,
, |IEF|,
element numbers IRBEF(i), IRBEF(i) + 1,
, IRBEF(i + 1)
− 1, of
B correspond to
the i-th effect.
B Vector of length NCOEF containing a least-squares solution
for the regression coefficients. (Input)
Here, if IEF>
0, then NCOEF =
IRBEF(IEF
+ 1) − 1;
and if IEF
≤ 0, then
NCOEF = INTCEP
− IEF.
If INTCEP
= 1, then B(1)
must be the estimated intercept.
R NCOEF by NCOEF upper triangular
matrix containing the R
matrix. (Input)
The R matrix can come from a regression
fit based on a QR decomposition of the matrix of regressors or based on a
Cholesky factorization RTR of the matrix of sums of squares
and crossproducts of the regressors. Elements to the right of a diagonal element
of R that is zero must
also be zero. A zero row indicates a nonfull rank model. For an R matrix that comes from a
regression fit with linear equality restrictions on the parameters, each row of
R corresponding to a
restriction must have a corresponding diagonal element that is negative. The
remaining rows of R must
have positive diagonal elements. Only the upper triangle of R is referenced.
DFE Degrees of freedom for error. (Input)
SSE Sum of squares for error. (Input)
AOV Vector of length 15 containing statistics relating to the analysis of variance. (Output)
I |
AOV(I) |
1 |
Degrees of freedom for regression |
2 |
Degrees of freedom for error |
3 |
Total degrees of freedom |
4 |
Sum of squares for regression |
5 |
Sum of squares for error |
6 |
Total sum of squares |
7 |
Regression mean square |
8 |
Error mean square |
9 |
F-statistic |
10 |
p-value |
11 |
R2 (in percent) |
12 |
Adjusted R2 (in percent) |
13 |
Estimated standard deviation of the model error |
14 |
Mean of the response (dependent) variable |
15 |
Coefficient of variation (in percent) |
If INTCEP = 1, the regression and total are corrected for the mean. If INTCEP = 0, the regression and total are not corrected for the mean, and AOV(14) and AOV(15) are set to NaN (not a number).
SQSS |IEF|
by 4 matrix containing in columns 1 through 4 the sequential degrees of freedom,
sum of squares, F-statistic, and
p-value. (Output)
Each row corresponds to an effect. If
IEF
= 0, SQSS is not
referenced and can be a vector of length one.
COEF NCOEF by 5 matrix
containing statistics relating to the regression coefficients.
(Output)
Each row corresponds to a coefficient in the model. Row INTCEP
+ I corresponds
to the coefficient for the I-th independent
variable. If INTCEP
= 1, the first row corresponds to the intercept. The statistics in the columns
are
Col. Description
1 Coefficient estimate.
2 Estimated standard error of the coefficient estimate.
3 t-statistic for the test that the coefficient is zero.
4 p-value for the two-sided t test.
5 Variance inflation factors. The square of the multiple correlation coefficient for the I-th regressor after all others can be obtained from COEF(I, 5) by the formula 1.0 − 1.0/coef(I, 5). If INTCEP = 0 or INTCEP = 1 and I = 1, the multiple correlation coefficient is not adjusted for the mean.
COVB NCOEF by NCOEF matrix that is
the estimated variance-covariance matrix of the estimated regression
coefficients when R is
nonsingular and is from an unrestricted regression fit. (Output)
See Comments for an explanation of COVB when R is singular or R is from a restricted regression
fit. If R is not needed,
COVB and R can share the same storage
locations.
INTCEP
Intercept option. (Input)
Default: INTCEP = 1.
INTCEP Action
0 An intercept is not in the model.
1 An intercept is in the model.
IEF Effect
option. (Input)
Default: IEF = 0.
The
absolute value of IEF is the number of
effects (sources of variation) in the model excluding the error. The sign of
IEF
specifies the following options:
IEF Meaning
< 0 Each effect corresponds to a single regressor (coefficient) in the model.
> 0 Each effect corresponds to one or more regressors. The association between the effects and the regressors is given by elements of IRBEF.
0 There are no effects in the model. INTCEP must equal 1.
LDR Leading
dimension of R
exactly as specified in the dimension statement in the calling
program. (Input)
Default: LDR = size (R,1).
PRINT Printing
option. (Input)
Default: PRINT = N.
PRINT is a character
string indicating what is to be printed. The PRINT string is
composed of one character print codes to control printing. These print codes are
given as follows:
PRINT(I : I) Printing that occurs
A All
N None
1 AOV
2 SQSS
3 COEF
4 COVB
The concatenated print codes A, N, 1, , 4 that comprise the PRINT string give the combination of statistics to be printed. Here are a few examples.
PRINT Printing Action
A All
N None
13 AOV and COEF
124 AOV, SQSS, and COVB
LDSQSS Leading
dimension of SQSS exactly as
specified in the dimension statement in the calling program.
(Input)
Default: LDSQSS = size (SQSS,1).
LDCOEF Leading
dimension of COEF exactly as
specified in the dimension statement in the calling program.
(Input)
Default: LDCOEF = size (COEF,1).
LDCOVB Leading
dimension of COVB exactly as specified
in the dimension statement in the calling program.
(Input)
Default: LDCOVB = size (COVB,1).
Generic: CALL RSTAT (IRBEF, B, R, DFE, SSE, AOV, SQSS, COEF, COVB [, ])
Specific: The specific interface names are S_RSTAT and D_RSTAT.
Single: CALL RSTAT (INTCEP, IEF, IRBEF, B, R, LDR, DFE, SSE, PRINT, AOV, SQSS, LDSQSS, COEF, LDCOEF, COVB, LDCOVB)
Double: The double precision name is DRSTAT.
Routine RSTAT
computes summary statistics from a fitted general linear model. The model is
y = X β + ɛ where y
is the n Χ 1 vector of
responses, X is the n Χ p matrix
of regressors, β is the
p Χ 1 vector of
regression coefficients, and ɛ is the
n Χ 1 vector of
errors whose elements are each independently distributed with mean 0 and
variance σ2. Routine RGIVN or routine RGLM can be used to compute the fit of the
model. Next, RSTAT
uses the results of this fit to compute summary statistics, including analysis
of variance, sequential sum of squares, t tests, and estimated
variance-covariance matrix of the estimated regression coefficients.
Some generalizations of the general linear model are allowed. If the i-th element of ɛ has variance σ2/wi and the weights wi are used in the fit of the model, RSTAT produces summary statistics from the weighted least-squares fit. More generally, if the variance-covariance matrix of ɛ is σ2V, RSTAT can be used to produce summary statistics from the generalized least-squares fit. (Routine RGIVN can be used to perform a generalized least-squares fit, by regressing y* on X* where y* = (T−1)Ty, X* = (T−1)TX and T satisfies TTT = V. Routines for computing y* and X* can be found in the IMSL MATH/LIBRARY.)
If the general linear model has the restriction H β = g on the regression parameters, and this restriction is used in the fit of the model by routine RLEQU, RSTAT produces summary statistics from this restricted least-squares fit.
The sequential sum of squares for the i-th regression parameter is given by
The regression sum of squares is given by the sum of the sequential sums of squares. If an intercept is in the model, the regression sum of squares is adjusted for the mean, i.e.,
is not included in the sum.
The estimate of σ2 is s2 (stored in AOV(8)) that is computed as SSE/DFE.
If R is nonsingular, the estimated variance-covariance matrix of
(stored in COVB) is computed by s2R−1(R−1)T.
If R
is singular, corresponding to rank (X) < p, a generalized
inverse is used. For a matrix G to be a gi(i = 1, 2, 3, or 4) inverse
of a matrix A, G must satisfy conditions j (for j
≤ i) for the
Moore-Penrose inverse but generally must fail conditions k (for k > i). The four conditions for
G to be a Moore-Penrose inverse of A are as follows:
1. AGA = A
2. GAG = G
3. AG is symmetric
4. GA is symmetric
In the case where R is singular, the method for obtaining COVB follows the discussion of Maindonald (1984, pages 101−103). Let Z be the diagonal matrix with diagonal elements defined by
Let G be the solution to RG = Z obtained by setting the i-th ({i : rii = 0}) row of G to zero. COVB is set to s2GGT. (G is a g3 inverse of R. For any g3 inverse of R, represented by
the result
is a symmetric g2 inverse of RTR = XTX. See Sallas and Lionti [1988].)
Note that COVB can only be used to get variances and covariances of estimable functions of the regression coefficients, i.e., nonestimable functions (linear combinations of the regression coefficients not in the space spanned by the nonzero rows of R) must not be used. See, for example, Maindonald (1984, pages 166−168) for a discussion of estimable functions.
The estimated standard errors of the estimated regression coefficients (stored in column 2 of COEF) are computed as square roots of the corresponding diagonal entries in COVB.
For the case where an intercept is in the model, put
equal to the matrix R with the first row and column deleted. Generally, the variance inflation factor (VIF) for the i-th regression coefficient is computed as the product of the i-th diagonal element of RTR and the i-th diagonal element of its computed inverse. If an intercept is in the model, the VIF for those coefficients not corresponding to the intercept uses the diagonal elements of
(see Maindonald 1984, page 40).
The preceding discussion can be modified to include the restricted least-squares problem. The modification is based on the work of Stirling (1981). Let the matrix D = diag(d1, d2, , dp) be a diagonal matrix with elements di = 0 if the i-th row of R corresponds to restriction. In the unrestricted case, D is simply the p Χ p identity matrix. The formula for COVB is s2GDGT. The formula for the sequential sum of squares for the i-th ({i : rii > 0}) regression parameter is given by
Sequential sums of squares for {i : rii ≤ 0} are set to zero.
For the restricted least-squares problem, the sequential and regression sums of squares correspond to those from a fitted reduced model obtained by first substituting the restriction H β = g into the model. In general, the reduced model is not unique. Care must be taken to interpret the sequential sums of squares in the context of the particular reduced model indicated by the R matrix. If g = 0, any of the reduced models that could be computed from the restrictions will produce the same regression sum of squares. However, if g ≠ 0, different reduced models resulting from the same restricted model can have different regressands, and hence, different total and regression sums of squares.
When R is nonsingular and comes from an
unrestricted regression fit, COVB is the estimated
variance-covariance matrix of the estimated regression coefficients,
and
COVB = (SSE/DFE) * (RTR)−1. Otherwise, variances and covariances of
estimable functions of the regression coefficients can be obtained using COVB,
and COVB = (SSE/DFE) * GDGT. Here, D
is the diagonal matrix with diagonal elements equal to 0 if the corresponding
rows of R are restrictions
and with diagonal elements equal to one otherwise. Also, G is a
particular generalized inverse of R. See the Description
section.
This example uses a data set discussed by Draper and Smith (1981, pages 629−630). This data set is put into the matrix X by routine GDATA (see Chapter 19, Utilities;). There are 4 independent variables and 1 dependent variable. Routine RGIVN is invoked to fit the regression model and RSTAT is invoked to compute summary statistics.
USE RSTAT_INT
USE GDATA_INT
USE RGIVN_INT
IMPLICIT NONE
! SPECIFICATIONS FOR LOCAL VARIABLES
INTEGER INTCEP, LDB, LDCOEF, LDCOVB, LDR, LDSCPE, LDSQSS, &
LDX, NCOEF, NDEP, NDX, NIND
PARAMETER (INTCEP=1, LDX=13, NDEP=1, NDX=5, NIND=4, &
LDSCPE=NDEP, LDSQSS=NIND, NCOEF=INTCEP+NIND, &
LDB=NCOEF, LDCOEF=NCOEF, LDCOVB=NCOEF, LDR=NCOEF)
!
INTEGER IDEP, IDO, IEF, IFRQ, IIND, INDDEP(1), INDIND(1), &
IRANK, IRBEF(1), IWT, NCOL, NRMISS, NROW
REAL AOV(15), B(LDB,NDEP), COEF(LDCOEF,5), &
COVB(LDCOVB,5), D(NCOEF), DFE, R(LDR,NCOEF), &
SCPE(LDSCPE,NDEP), SQSS(LDSQSS,4), SSE, TOL, &
X(LDX,NDX), XMAX(NCOEF), XMIN(NCOEF)
CHARACTER PRINT*5
!
CALL GDATA (5, X, NROW, NCOL)
IIND = -NIND
IDEP = -NDEP
CALL RGIVN (X, IIND, INDIND, IDEP, INDDEP, B, R=R, DFE=DFE, &
SCPE=SCPE)
PRINT = 'A'
IEF = -NIND
SSE = SCPE(1,1)
!
CALL RSTAT (IRBEF, B(:, 1), R, DFE, SSE, AOV, SQSS, COEF, COVB, &
IEF=IEF, PRINT=PRINT)
!
END
R-squared Adjusted Est. Std.
Dev.
Coefficient of
(percent) R-squared of Model
Error Mean Var.
(percent)
98.238
97.356
2.446
95.42
2.563
* * * Analysis of Variance * *
*
Sum of
Mean
Prob.
of
Source
DF Squares Square
Overall F Larger
F
Regression
4 2667.9
667.0 111.479
0.0000
Residual
8
47.9 6.0
Corrected
Total 12
2715.8
* * * Sequential
Statistics * * *
Indep. Degrees of
Sum
of
Prob. of
Variable Freedom
Squares F-statistic Larger
F
1
1 1450.1
242.368 0.0000
2
1 1207.8
201.870 0.0000
3
1
9.8 1.637
0.2366
4
1
0.2 0.041
0.8441
* * * Inference on Coefficients * *
*
Standard
Prob. of Variance
Coef.
Estimate Error t-statistic
Larger |t| Inflation
1 62.41
70.07
0.891 0.3991
10668.5
2
1.55
0.74 2.083
0.0708
38.5
3
0.51
0.72
0.705 0.5009
254.4
4
0.10
0.75
0.135
0.8959 46.9
5
-0.14
0.71 -0.203
0.8441 282.5
* * *
Variance-Covariance Matrix for the Coefficient Estimates * *
*
1
2
3
4
5
1
4909.95
-50.51
-50.60
-51.66
-49.60
2
0.55
0.51
0.55
0.51
3
0.52
0.53
0.51
4
0.57
0.52
5
0.50
A one-way analysis of covariance model is fitted to the turkey data discussed by Draper and Smith (1981, pages 243−249). The response variable is turkey weight y (in pounds). Three groups of turkeys corresponding to the three states where they were reared are used. The age of a turkey (in weeks) is the covariate. The explanatory variables are age, group, and interaction. The model is
yij = μ + βxij + α i + βixij + ɛ ij i = 1, 2, 3; j = 1, 2, , ni
where α3 = 0 and β3 = 0. Routine RGLM is used to fit the model with the option IDUMMY = 2. Then, RSTAT is used to compute summary statistics. The fitted model gives three separate lines with slopes 0.506, 0.470, and 0.445. The F test for interaction (the last effect) suggests omitting the interaction from the model and using a model with identical slopes for each group.
USE RSTAT_INT
USE RGLM_INT
IMPLICIT NONE
! SPECIFICATIONS FOR PARAMETERS
INTEGER IDEP, IEF, INTCEP, LDB, LDCOEF, LDCOVB, LDR, LDSCPE, &
LDSQSS, LDX, MAXB, MAXCL, NCLVAR, NCOL, NROW, J
PARAMETER (IDEP=1, IEF=3, INTCEP=1, LDX=13, MAXB=6, MAXCL=3, &
NCLVAR=1, NCOL=3, NROW=13, LDB=MAXB, LDCOEF=MAXB, &
LDCOVB=MAXB, LDR=MAXB, LDSCPE=IDEP, LDSQSS=IEF)
!
INTEGER IDO, IDUMMY, IFRQ, INDCL(NCLVAR), INDDEP(IDEP), &
INDEF(4), IRANK, IRBEF(IEF+1), IWT, NCLVAL(NCLVAR), &
NRMISS, NVEF(IEF)
REAL AOV(15), B(LDB,IDEP), CLVAL(MAXCL), &
COEF(LDCOEF,5), COVB(LDCOVB,MAXB), D(MAXB), DFE, &
R(LDR,MAXB), SCPE(LDSCPE,IDEP), SQSS(LDSQSS,4), SSE, &
TOL, X(LDX,NCOL), XMAX(MAXB), XMIN(MAXB)
CHARACTER PRINT*1
!
DATA (X(1,J),J=1,3)/25, 13.8, 3/
DATA (X(2,J),J=1,3)/28, 13.3, 1/
DATA (X(3,J),J=1,3)/20, 8.9, 1/
DATA (X(4,J),J=1,3)/32, 15.1, 1/
DATA (X(5,J),J=1,3)/22, 10.4, 1/
DATA (X(6,J),J=1,3)/29, 13.1, 2/
DATA (X(7,J),J=1,3)/27, 12.4, 2/
DATA (X(8,J),J=1,3)/28, 13.2, 2/
DATA (X(9,J),J=1,3)/26, 11.8, 2/
DATA (X(10,J),J=1,3)/21, 11.5, 3/
DATA (X(11,J),J=1,3)/27, 14.2, 3/
DATA (X(12,J),J=1,3)/29, 15.4, 3/
DATA (X(13,J),J=1,3)/23, 13.1, 3/
DATA INDCL/3/, NVEF/1, 1, 2/, INDEF/1, 3, 1, 3/, INDDEP/2/
!
IDUMMY = 2
CALL RGLM (X, INDCL, NVEF, INDEF, IDEP, INDDEP, MAXCL, &
B, IDUMMY=IDUMMY, IRBEF=IRBEF, R=R, DFE=DFE, SCPE=SCPE)
!
SSE = SCPE(1,1)
PRINT = 'A'
CALL RSTAT (IRBEF, B(:,1), R, DFE, SSE, AOV, SQSS, COEF, COVB, &
ief=ief,PRINT=PRINT)
!
END
R-squared Adjusted Est. Std.
Dev.
Coefficient of
(percent) R-squared of Model
Error Mean Var.
(percent)
98.208
96.929
0.3176
12.78
2.484
* * * Analysis of Variance * *
*
Sum of
Mean
Prob.
of
Source
DF Squares Square
Overall F Larger
F
Regression
5 38.71
7.742 76.744
0.0000
Residual
7
0.71 0.101
Corrected
Total 12
39.42
* * * Sequential Statistics * *
*
Degrees
of Sum
of
Prob. of
Effect Freedom
Squares F-statistic Larger F
1
1 26.20
259.728 0.0000
2
2 12.40
61.477 0.0000
3
2
0.11 0.520
0.6156
* * * Inference on Coefficients * *
*
Standard
Prob. of Variance
Coef.
Estimate Error t-statistic
Larger |t| Inflation
1 2.475
1.264
1.959 0.0910
205.7
2
0.445
0.050
8.861
0.0000 3.8
3 -3.454
1.531 -2.257
0.0586 64.3
4 -2.775
4.109 -0.675
0.5211 463.4
5 0.061
0.060
1.013
0.3447 68.1
6 0.025
0.151
0.166 0.8729
472.3
* * * Variance-Covariance Matrix for the
Coefficient Estimates * *
*
1
2
3
4
5
1
1.5965
-0.0631
-1.5965
-1.5965
0.0631
2
0.0025
0.0631
0.0631
-0.0025
3
2.3425
1.5965
-0.0913
4
16.8801
-0.0631
5
0.0036
6
1
0.0631
2
-0.0025
3
-0.0631
4
-0.6179
5
0.0025
6 0.0227
A two-way analysis-of-variance model is fitted to balanced data discussed by Snedecor and Cochran (1967, Table 12.5.1, page 347). The responses are the weight gains (in grams) of rats fed diets varying in two componentslevel of protein and source of protein. The model is
yijk = μ + α i + βj + γij + ɛ ijk i = 1, 2; j = 1, 2, 3; k = 1, 2, , 10
where
Routine RGLM is used to fit the model with the IDUMMY = 0 option. Then, RSTAT is used to compute summary statistics.
USE RSTAT_INT
USE RGLM_INT
IMPLICIT NONE
INTEGER IDEP, IEF, LDB, LDCOEF, LDCOVB, LDR, LDSCPE, LDSQSS, &
LDX, LINDEF, MAXB, MAXCL, NCLVAR, NCOL, NEF, NROW
PARAMETER (IDEP=1, LINDEF=4, MAXB=12, MAXCL=5, NCLVAR=2, &
NCOL=3, NEF=3, NROW=60, IEF=NEF, LDB=MAXB, &
LDCOEF=MAXB, LDCOVB=MAXB, LDR=MAXB, LDSCPE=IDEP, &
LDSQSS=NEF, LDX=NROW)
!
INTEGER IDO, IDUMMY, IFRQ, INDCL(NCLVAR), INDDEP(IDEP),&
INDEF(LINDEF), INTCEP, IRANK, IRBEF(NEF+1), IWT, &
NCLVAL(NCLVAR), NRMISS, NVEF(NEF)
REAL AOV(15), B(LDB,IDEP), CLVAL(MAXCL), &
COEF(LDCOEF,5), COVB(LDCOVB,MAXB), D(MAXB), DFE, &
R(LDR,MAXB), SCPE(LDSCPE,IDEP), SQSS(LDSQSS,4), SSE, &
TOL, X(LDX,NCOL), XMAX(MAXB), XMIN(MAXB)
CHARACTER PRINT*1
!
DATA X/73.0, 102.0, 118.0, 104.0, 81.0, 107.0, 100.0, 87.0, &
117.0, 111.0, 98.0, 74.0, 56.0, 111.0, 95.0, 88.0, 82.0, &
77.0, 86.0, 92.0, 94.0, 79.0, 96.0, 98.0, 102.0, 102.0, &
108.0, 91.0, 120.0, 105.0, 90.0, 76.0, 90.0, 64.0, 86.0, &
51.0, 72.0, 90.0, 95.0, 78.0, 107.0, 95.0, 97.0, 80.0, &
98.0, 74.0, 74.0, 67.0, 89.0, 58.0, 49.0, 82.0, 73.0, 86.0, &
81.0, 97.0, 106.0, 70.0, 61.0, 82.0, 30*1.0, 30*2.0, &
10*1.0, 10*2.0, 10*3.0, 10*1.0, 10*2.0, 10*3.0/
DATA INDCL/2, 3/, NVEF/1, 1, 2/, INDEF/2, 3, 2, 3/, INDDEP/1/
!
IDUMMY = 0
CALL RGLM (X, INDCL, NVEF, INDEF, IDEP, INDDEP, MAXCL, B, &
IDUMMY=IDUMMY, IRBEF=IRBEF, R=R, DFE=DFE, SCPE=SCPE)
!
SSE = SCPE(1,1)
PRINT = 'A'
CALL RSTAT (IRBEF, B(:,1), R, DFE, SSE, AOV, SQSS, COEF, &
COVB, IEF=IEF, PRINT=PRINT)
!
END
R-squared Adjusted Est. Std.
Dev.
Coefficient of
(percent) R-squared of Model
Error Mean Var.
(percent)
28.477
21.854
14.65
87.87
16.67
* * *
Analysis of Variance * *
*
Sum of
Mean
Prob.
of
Source
DF Squares Square
Overall F Larger
F
Regression
5 4612.9
922.6 4.300
0.0023
Residual
54 11586.0
214.6
Reduced Model Total
59
16198.9
* * * Sequential
Statistics * * *
Degrees
of Sum
of
Prob. of
Effect Freedom
Squares F-statistic Larger F
1
1 3168.3
14.767 0.0003
2
2
266.5 0.621
0.5411
3
2 1178.1
2.746
0.0732
* * * Inference on Coefficients * *
*
Standard
Prob.
of Variance
Coef.
Estimate Error t-statistic
Larger |t| Inflation
1 87.87
1.891
46.47 0.0000
1.000
2
7.27
1.891
3.84
0.0003 NaN
3 -7.27
1.891
-3.84 0.0003
1.000
4
1.73
2.674
0.65
0.5196 NaN
5 -2.97
2.674
-1.11 0.2722
1.333
6
1.23
2.674
0.46 0.6465
1.333
7
3.13
2.674
1.17
0.2465 NaN
8 -6.27
2.674
-2.34
0.0228 NaN
9
3.13
2.674
1.17
0.2465 NaN
10 -3.13
2.674
-1.17
0.2465 NaN
11
6.27
2.674
2.34 0.0228
1.333
12
-3.13
2.674
-1.17 0.2465
1.333
* * * Variance-Covariance Matrix for
the Coefficient Estimates * *
*
1
2
3
4
5
1
3.57593
0.00000
0.00000
0.00000
0.00000
2
3.57593
-3.57593
0.00000
0.00000
3
3.57593
0.00000
0.00000
4
7.15185
-3.57592
5
7.15185
6
7
8
9
10
1
0.00000
0.00000
0.00000
0.00000
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