wls
Syntax
wls(Y, X, W, [intercept=true], [mode=0])
Arguments
Y is a dependent variable.
X is an independent variable.
Y is a vector. X can be a matrix, table or tuple. When X is a matrix, if the number of rows is equal to the length of Y, each column of X is a factor. If the number of rows is not equal to the length of Y, but the number of columns is equal to the length of Y, each row of X is a factor.
W is a vector indicating the weight in which each element is a non-negative.
intercept is a boolean variable that indicates whether to include the intercept in regression. The default value is true. When it is true, the system automatically adds a column of “1” to X to generate the intercept.
mode is an integer that could be 0, 1, or 2.
0: a vector of the coefficient estimates
1: a table with coefficient estimates, standard error, t-statistics, and p-value
2: a dictionary with all statistics
ANOVA (one-way analysis of variance)
Source of Variance |
DF (degree of freedom) |
SS (sum of square) |
MS (mean of square) |
F (F-score) |
Significance |
|---|---|---|---|---|---|
Regression |
p |
sum of squares regression, SSR |
regression mean square, MSR=SSR/R |
MSR/MSE |
p-value |
Residual |
n-p-1 |
sum of squares error, SSE |
mean square error, MSE=MSE/E |
||
Total |
n-1 |
sum of squares total, SST |
RegressionStat (Regression statistics)
Item |
Description |
|---|---|
R2 |
R-squared |
AdjustedR2 |
The adjusted R-squared corrected based on the degrees of freedom by comparing the sample size to the number of terms in the regression model. |
StdError |
The residual standard error/deviation corrected based on the degrees of freedom. |
Observations |
The sample size. |
Coefficient
Item |
Description |
|---|---|
factor |
Independent variables |
beta |
Estimated regression coefficients |
StdError |
Standard error of the regression coefficients |
tstat |
t statistic, indicating the significance of the regression coefficients |
Residual: the difference between each predicted value and the actual value.
Details
Return the result of an weighted-least-squares regression of Y on X.
Examples
$ x1=1 3 5 7 11 16 23
$ x2=2 8 11 34 56 54 100
$ y=0.1 4.2 5.6 8.8 22.1 35.6 77.2;
$ w=rand(10,7)
$ wls(y, x1, w)
[-17.6177 4.0016]
$ wls(y, (x1,x2), w);
[-17.4168 3.0481 0.2214]
$ wls(y, (x1,x2), w, 1, 1);
factor |
beta |
stdError |
tstat |
pvalue |
|---|---|---|---|---|
Intercept |
-17.4168 |
4.8271 |
-3.6081 |
0.0226 |
x1 |
3.0481 |
1.6232 |
1.8779 |
0.1336 |
x2 |
0.2214 |
0.3699 |
0.5986 |
0.5817 |
$ wls(y, (x1,x2), w,1, 2);
Coefficient->
factor beta stdError tstat pvalue
--------- --------- -------- --------- --------
intercept -10.11392 4.866583 -2.078239 0.106234
x1 3.938138 2.061191 1.910613 0.128655
x2 -0.088542 0.446667 -0.198227 0.852534
Residual->[6.452866,3.207839,-3.002812,-5.642629,-6.147264,-12.515038,5.590914]
RegressionStat->
item statistics
------------ ----------
R2 0.957998
AdjustedR2 0.936997
StdError 17.172833
Observations 7
ANOVA->
Breakdown DF SS MS F Significance
---------- -- ------------ ------------ --------- ------------
Regression 2 26905.306594 13452.653297 45.616718 0.001764
Residual 4 1179.624835 294.906209
Total 6 28084.931429
$ x=matrix(1 4 8 2 3, 1 4 2 3 8, 1 5 1 1 5);
$ w=rand(8,5)
$ wls(1..5, x,w,0,1);
factor |
beta |
stdError |
tstat |
pvalue |
|---|---|---|---|---|
beta0 |
0.0026 |
1.4356 |
0.0018 |
0.9988 |
beta1 |
-1 |
1.2105 |
-0.8261 |
0.5605 |
beta2 |
0.4511 |
0.5949 |
0.7582 |
0.587 |
beta3 |
1.687 |
1.7389 |
0.9701 |
0.5097 |