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 Principal Component(PC) 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.
PC Regression circumvents this by decomposing X into orthogonal scores and loadings .
In this method, we approximate the matrix by the first Principal Components (PCs), usually obtained from the singular value decomposition(SVD):

Next, we regress on the scores, which leads to regression coefficients

Path of PC Regression:

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

A. PC Regression window:

After opening the PC 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 select 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 PC 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 . The 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 “PCRResid”. This message will appear if the residuals are saved successfully:
“Name of Residual Variable Residuals added in Data Table.”
For example, “PCRResid Residuals added in Data Table.”.