Example Variables(Open Data):

The Fennema-Sherman Mathematics Attitude Scales (FSMAS) are among the most popular instruments used in studies of attitudes toward mathematics. FSMAS contains 36 items. Also, scales of FSMAS have Confidence, Effectance Motivation, and Anxiety. The sample includes 425 teachers who answered all 36 items. In addition, other characteristics of teachers, such as their age, are included in the data.
You can select your data as follows:
1-File
2-Open data
(See Open Data)

The data is stored under the name FSMAS-T(You can download this data from here ).
You can edit the imported data via the following path:
1-File
2-Edit Data
(See Edit Data)

Example Variables(Compute Variable):

The three variables of Confidence, Effectance Motivation, and Anxiety can be calculated through the following path:
1-Transform
2-Compute Variable
Items starting with the letters C, M, and A are related to the variables Confidence, Effectance Motivation, and Anxiety, respectively.
(See Compute Variable)

Introduction to Partial Least Squares (PLS) Regression

The multiple linear regression can be written in matrix notation as

Where y and are vectors of the values of the response variable and the errors for the various observations, and is a matrix.
In the multiple linear regression, the least-squares solution for is given by .
The problem often is that is singular, either because the number of variables (columns) in exceeds the number of objects (rows), or because of collinearities.
PLS Regression circumvent this by decomposing X into orthogonal scores and loadings
In this method, the components, called Latent Variables (LVs) in this context, are obtained iteratively. One starts with the SVD(singular value decomposition) of the cross product matrix , thereby including information on variation in both and , and on the correlation between them. The first left and right singular vectors, and , are used as weight vectors for and , respectively, to obtain scores and :


Where and are initialised as and , respectively. The scores are often normalised:

The scores are not actually necessary in the regression but are often saved for interpretation purposes. Next, and loadings are obtained by regressing against the same vector :

Finally, the data matrices are deflated: the information related to this latent variable, in the form of the outer products and , is subtracted from the (current) data matrices and .


The estimation of the next component can then start from the SVD of the cross-product matrix . After every iteration, vectors , , and are saved as columns in matrices , , and , respectively. One complication is that columns of matrix can not be compared directly: they are derived from successively deflated matrices and . It has been shown that an alternative way to represent the weights, in such a way that all columns relate to the original matrix, is given by
Now, instead of regressing on , we use scores to calculate the regression coefficients, and later convert these back to the realm of the original variables by pre-multiplying with matrix (since ):

Path of PLS Regression:

You can perform PLS regression by the following path:
1-Exploratory Analysis
2- Regression
3-PLS

A. PLS Regression window:

After opening the PLS Regression window, you can start the model analysis process.

B. Select Dependent Variable:

You can select the dependent variable through this button. After opening the window, you can choose it by selecting the desired variable.
For example, the variable Confidence is selected in this data.

C. Select Independent Variables:

You can select the independent variables through this button. After the window opens, you can select them by selecting the desired variables.
For example, the variables Effectance Motivation and Anxiety are selected in this data.

D. Run Regression:

You can see the results of the PLS regression in the main window by clicking this button.
Results include the following:
-Parameter Estimates
-Model Summary
-Collinearity Diagnostics

E. Results:

Parameters are estimated based on the jack-knife method.
At first, all and variables are regarded as active. Then the following process was iterated automatically until each apparently useless and variable had been detected as non-significant and made passive. The full leave-one-product-out cross-validation was applied for the remaining active and variables, and the optimal rank was determined. The approximate uncertainty variance of the PC regression coefficients was then estimated by jack-knifing:

Where estimated uncertainty variance of
the regression coefficient at the cross-validated rank using all the objects
the regression coefficient at the rank using all objects except the object left out in cross-validation segment

As a rough significance test a t-test (n degrees of freedom) was performed for each element in relative to the square root of its estimated uncertainty variance , and the optimum significance level was noted for each variable and for each variable.

*


*MSEP:


*RMSEP:


* Collinearity Diagnostics
In the Collinearity Diagnostics table, the results of multicollinearity in a set of multiple regression variables are given.
For each independent variable, the VIF index is calculated in two steps:
STEP1:
First, we run an ordinary least square regression that has x_i as a function of all the other explanatory variables in the first equation.
STEP2:
Then, calculate the VIF factor with the following equation:
Where is the coefficient of determination of the regression equation in step one. Also,
A rule of decision making is that if then multicollinearity is high (a cutoff of 5 is also commonly used)

F. Save Regression:

By clicking this button, you can save the regression results. After opening the save results window, you can save the results in “text” or “Microsoft Word” format.

G. Add Residuals & Name of Residual Variable:

By clicking on this button, you can save the residuals of the regression model with the desired name(Name of Residual Variable). The default software for the residual names is “PLSResid”. This message will appear if the residuals are saved successfully:
“Name of Residual Variable Residuals added in Data Table.”
For example, “PLSResid Residuals added in Data Table.”.