# 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

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
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