Hierarchical Multiple Regression SPSS

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In this tutorial, we will learn how to perform hierarchical multiple regression analysis SPSS, which is a variant of the basic multiple regression analysis that allows specifying a fixed order of entry for variables (regressors) to control for the effects of covariates or to test the effects of certain predictors independent of the influence of other.

Step By Step Procedure of Hierarchical Multiple Regression SPSS

The basic command for hierarchical multiple regression analysis SPSS is “regression -> linear”:

Hierarchical Multiple Regression SPSS

In the main dialog box of linear regression (as given below), input the dependent variable. For example “income” variable from the sample file of customer_dbase.sav available in the SPSS installation directory.

Next, enter a set of predictor variables into an independent(s) pan. These variables that you want SPSS to put into the regression model first (that you want to control for when testing the variables). For example, in this analysis, we want to find out whether the “Number of people in the house” predicts the “Household income in thousands”.

We are also concerned that other variables like age, education, gender, union member, or retirement might be associated with both the “number of people in the house” and “household income in thousands”. To make sure that these variables (age, education, gender, union member, and retired) do not explain away the entire association between the “number of people in the house” and “Household income in thousands”, let’s put them into the model first.

This ensures that they will get credit for any shared variability that they may have with the predictor that we are interested in, “Number of people in the house”. any observed effect of “Number of people in the house” can then be said to be “independent of the effects of these variables that already have been controlled for. See the figure below

Linear Regression Variable

In the next step put the variable that we are interested in, which is the “number of people in the house”. To include it in the model click the “NEXT” button. You will see all of the predictors (that were entered previously) disappear. Note that they are still in the model, just not on the current screen (block). You will also see Block 2 of 2 above the “independent(s)” pan.

Hierarchical Regression

Now click the “OK” button to run the analysis.

Note you can also hit the “NEXT” button again if you are interested in entering a third or fourth (and so on) block of variables.

Often researchers enter variables as related sets. For example demographic variables in the first step, all potentially confounding variables in the second step, and then the variables that you are most interested in in the third step. However, it is not necessary to follow. One can also enter each variable as a separate step if that seems more logical based on the design of your experiment.

Output Hierarchical Multiple Regression Analysis

Using just the default “Enter” method, with all the variables in Block 1 (demographics) entered together, followed by “number of people in the house” as a predictor in Block 2, we get the following output:

Output Hierarchical Regression

The first table of output windows confirms that variables are entered in each step.

The summary table shows the percentage of explained variation in the dependent variable that can be accounted for by all the predictors together. The change in $R^2$ (R-squared) is a way to evaluate how much predictive power was added to the model by the addition of another variable in STEP 2. In our example, predictive power does not improve with the addition of another predictor in STEP 2.

Hierarchical Regression Output

The overall significance of the model can be checked from this ANOVA table. In this case, both models are statistically significant.

Hierarchical Regression Output

The coefficient table is used to check the individual significance of predictors. For model 2, the Number of people in the household is statistically non-significant, therefore excluded from the model.

Learn about Multiple Regression Analysis

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Learn Cholesky Transformation (2020)

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Given the covariances between variables, one can write an invertible linear transformation that “uncorrelated” the variables. Contrariwise, one can transform a set of uncorrelated variables into variables with given covariances. This transformation is called Cholesky Transformation; represented by a matrix that is the “Square Root” of the covariance matrix.

The Square Root Matrix

Given a covariance matrix $\Sigma$, it can be factored uniquely into a product $\Sigma=U’U$, where $U$ is an upper triangle matrix with positive diagonal entries. The matrix $U$ is the Cholesky (or square root) matrix. If one prefers to work with the lower triangular matrix entries ($L$), then one can define $$L=U’ \Rightarrow \quad \Sigma = LL’.$$

This is the form of the Cholesky decomposition given by Golub and Van Lean in 1996. They provided proof of the Cholesky Decomposition and various ways to compute it.

The Cholesky matrix transforms uncorrelated variables into variables whose variances and covariances are given by $\Sigma$. If one generates standard normal variates, the Cholesky transformation maps the variables into variables for the multivariate normal distribution with covariance matrix $\Sigma$ and centered at the origin (%MVN(0, \Sigma)$).

Generally, pseudo-random numbers are used to generate two variables sampled from a population with a given degree of correlation. Property is used for a set of variables (correlated or uncorrelated) in the population, a given correlation matrix can be imposed by post-multiplying the data matrix $X$ by the upper triangular Cholesky Decomposition of the correlation matrix R. That is

  • Create two variables using the pseudo-random number, let the names be $X$ and $Y$
  • Create the desired correlation matrix between variables using $Y=X*R + Y*\sqrt{1-r^2},$
    where $r$ is the desired correlation value. $X$ and $Y$ variables will have an exact desired relationship between them. For a larger number of times, the distribution of correlation will be centered on $r$.

The Cholesky Transformation: The Simple Case

Suppose you want to generate multivariate normal data that are uncorrelated but have non-unit variance. The covariance matrix is the diagonal matrix of variance: $\Sigma = diag(\sigma_1^2,\sigma_2^2,\cdots, \sigma_p^2)$. The $\sqrt{\Sigma}$ is the diagnoal matrix $D$ that consists of the standard deviations $\Sigma = D’D$, where $D=diag(\sigma_1,\sigma_2,\cdots, \sigma_p)$.

Geometrically, the $D$ matrix scales each coordinate direction independent of other directions. The $X$-axix is scaled by a factor of 3, whereas the $Y$-axis is unchanged (scale factor of 1). The transformation $D$ is $diag(3,1)$, which corresponds to a covariance matrix of $diag(9,1)$.

Thinking the circles in Figure ‘a’ as probability contours for multivariate distribution $MNV(0, I)$, and Figure ‘b’ as the corresponding probability ellipses for the distribution $MNV(0, D)$.

Cholesky Transformation
# define the correlation matrix
C <- matrix(c(1.0, 0.6, 0.3,0.6, 1.0, 0.5,0.3, 0.5, 1.0),3,3)

# Find its cholesky decomposition
U = chol(C)

#generate correlated random numbers from uncorrelated
#numbers by multiplying them with the Cholesky matrix.
x <- matrix(rnorm(3000),1000,3)
xcorr <- x%*%U
cor(xcorr)
Cholesky Transformation
https://itfeature.com

Reference: Cholesky Transformation to correlate and Uncorrelated variables

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MCQs General Knowledge

What is research? Why do we conduct it?

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An important question about discovering some new knowledge is What is Research? Why do we do Research? The answer of What is research and how it is conducted is explained below.

What is Research

Research is an inquiry. It is a process of discovering some new knowledge, that involves multiple elements such as theory development and testing, empirical inquiry, and sharing the generated knowledge with others such as experts and colleagues. A short description of the elements of theory is:

The theory is a set of ideas and perceptions that helps people to understand complex concepts and the relationships among these concepts. To develop and/or test a theory, researchers conduct empirical inquiries, collect and analyze relevant data, and discuss the findings from empirical results. Once theories have been through the research process, it is necessary to share the results of the studies with others such as researchers (related to the study) present papers at conferences, and publish reports in journals and other publications.

There are two ways to use the results of a study:

  1. The results may contribute to researchers’ general understanding of the topic they have researched i.e. studied and may contribute to, understanding how the economy works, why price inflation happens, which factors increase a candidate’s chances of winning an election, etc. The generalizations of results that researchers draw from their studies on these issues can be shared with other researchers and the general public to advance society for the understanding of the topic.
  2. The results of a study may contribute to solving particular problems in a nation, state, or community. For example, a study on the healthcare needs of the elderly in a community may discover that their primary need is finding vehicles for transportation when they want to visit their doctors. The leaders of the community (such as the mayor, and city council) may use this information from the healthcare study, to allocate some money for the transportation needs of the elderly in the next year’s budget.

Therefore, research is a tool that builds blocks of knowledge that in turn contribute to the development of science.

What is Research, Why we Conduct a Research?

Why conduct research?

  • To understand a phenomenon, situation, or behavior under study.
  • To test existing theories and to develop new theories based on existing ones.
  • To answer different questions of “how”, “what”, “which”, “when” and “why” about a phenomenon, behavior, or situation.
  • Research-related activities contribute to forming (making) new knowledge and expanding the existing knowledge base.

High-Quality Research

Nowadays one can collect/ gather information about almost anything from the Internet Just do a Google search. But a question is, does every Google search good research? Not quite! Do remember, though you will find some of the information, it may or may not be valid or high-quality information. A lot of the information available on the Internet is good and useful, but some are not. There may be misinformation too on the Internet. The information you find on the internet may be someone’s pure opinion, have some fabrication in it, or be based on some unsystematic research or unauthentic information. In short, the information may be valid (objective, true).

Therefore, a high-quality research project:

  • is based on the scholarly work that has been already done by others in the field,
  • can be replicated/ reproduced,
  • is a generalization to other settings,
  • is based on some logical rationale and tied to other existing theory;
  • is doable and can be done practically, i.e. when deciding the scope of research. A researcher should consider the availability of time and resources,
  • generates some new questions,
  • is incremental,
  • is an apolitical (politically neutral) activity that should be undertaken for the betterment of society.

Two Types/ Purposes

Typically, there are two types/purposes: Basic Research and Applied Research

  1. To find out about truths regarding human behaviors, societies, economy, etc., or to understand them better. This type is called basic research.
  2. To answer practical questions and support making informed decisions. This type is called applied research.

Note that, most of the public administration and public policy research projects are of the second kind.

Learn about Qualitative vs Quantitative Research

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Model Selection Criteria (2019)

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All models are wrong, but some are useful. Model selection criteria are rules used to select a (statistical) model among competing models, based on given data.

Several model selection criteria are used to choose among a set of candidate models, and/ or compare models for forecasting purposes.

All model selection criteria aim at minimizing the residual sum of squares (or increasing the coefficient of determination value). The criterion Adj-$R^2$, Akaike Information, Bayesian Information Criterion, Schwarz Information Criterion, and Mallow’s $C_p$ impose a penalty for including an increasingly large number of regressors. Therefore, there is a trade-off between the goodness of fit of the model and its complexity. The complexity refers to the number of parameters in the model.

Model Selection Criteria

Model Selection Criteria: Coefficient of Determination ($R^2$)

$$R^2=\frac{\text{Explained Sum of Square}}{\text{Total Sum of Squares}}=1-\frac{\text{Residuals Sum of Squares}}{\text{Total Sum of Squares}}$$

Adding more variables to the model may increase $R^2$ but it may also increase the variance of forecast error.
There are some problems with $R^2$

  • It measures in-sample goodness of fit (how close an estimated $Y$ value is to its actual values) in the given sample. There is no guarantee that $R^2$ will forecast well out-of-sample observations.
  • In comparing two or more $R^2$’s, the dependent variable must be the same.
  • $R^2$ cannot fall when more variables are added to the model.

Model Selection Criteria: Adjusted Coefficient of Determination ($R^2$)

$$\overline{R}^2=1-\frac{RSS/(n-k}{TSS(n-1)}$$

$\overline{R}^2 \ge R^2$ shows that the adjusted $R^2$ penalizes for adding more regressors (explanatory variables). Unlike $R^2$, the adjusted $R^2$ will increase only if the absolute $t$-value of the added variable is greater than 1. For comparative purposes, $\overline{R}^2$ is a better measure than $R^2$. The regressand (dependent variable) must be the same for the comparison of models to be valid.

Model Selection Criteria: Akaike’s Information Criterion (AIC)

$$AIC=e^{\frac{2K}{n}}\frac{\sum \hat{u}^2_i}{n}=e^{\frac{2k}{n}}\frac{RSS}{n}$$
where $k$ is the number of regressors including the intercept. The formula of AIC is

$$\ln AIC = \left(\frac{2k}{n}\right) + \ln \left(\frac{RSS}{n}\right)$$
where $\ln AIC$ is natural log of AIC and $\frac{2k}{n}$ is penalty factor.

AIC imposes a harsher penalty than the adjusted coefficient of determination for adding more regressors. In comparing two or more models, the model with the lowest value of AIC is preferred. AIC is useful for both in-sample and out-of-sample forecasting performance of a regression model. AIC is used to determine the lag length in an AR(p) model also.

Model Selection Criteria: Schwarz’s Information Criterion (SIC)

\begin{align*}
SIC &=n^{\frac{k}{n}}\frac{\sum \hat{u}_i^2}{n}=n^{\frac{k}{n}}\frac{RSS}{n}\\
\ln SIC &= \frac{k}{n} \ln n + \ln \left(\frac{RSS}{n}\right)
\end{align*}
where $\frac{k}{n}\ln\,n$ is the penalty factor. SIC imposes a harsher penalty than AIC.

Like AIC, SIC is used to compare the in-sample or out-of-sample forecasting performance of a model. The lower the values of SIC, the better the model.

Model Selection Criteria: Mallow’s $C_p$ Criterion

For Model selection the Mallow criteria is
$$C_p=\frac{RSS_p}{\hat{\sigma}^2}-(n-2p)$$
where $RSS_p$ is the residual sum of the square using the $p$ regression in the model.
\begin{align*}
E(RSS_p)&=(n-p)\sigma^2\\
E(C_p)&\approx \frac{(n-p)\sigma^2}{\sigma^2}-(n-2p)\approx p
\end{align*}
A model that has a low $C_p$ value, about equal to $p$ is preferable.

Model Selection Criteria: Bayesian Information Criteria (BIC)

The Bayesian information Criteria is based on the likelihood function and it is closely related to the AIC. The penalty term in BIC is larger than in AIC.
$$BIC=\ln(n)k-2\ln(\hat{L})$$
where $\hat{L}$ is the maximized value of the likelihood function of the regression model.

Cross-Validation

Cross-validation is a technique where the data is split into training and testing sets. The model is trained on the training data and then evaluated on the unseen testing data. This helps assess how well the model generalizes to unseen data and avoids overfitting.

Note that no one of these criteria is necessarily superior to the others.

Read more about Correlation and Regression Analysis

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