A large bank wants to gain insight into their employees’ job satisfaction. They carried out a survey, the results of which are in bank_clean.sav. The survey included some statements regarding job satisfaction, some of which are shown below.

## Research Question

The main research question for today is
which factors contribute (most) to overall job satisfaction?
as measured by overall (“I'm happy with my job”). The usual approach for answering this is *predicting* job satisfaction from these factors with **multiple linear regression analysis**.^{2,6} This tutorial will explain and demonstrate each step involved and we encourage you to run these steps yourself by downloading the data file.

## Data Check 1 - Coding

One of the best SPSS practices is making sure you've an idea of what's in your data before running any analyses on them. Our analysis will use overall through q9 and their variable labels tell us what they mean. Now, if we look at these variables in data view, we see they contain values 1 through 11.

So what do these values mean and -importantly- is this the same for all variables? A great way to find out is running the syntax below.

***Check coding: higher values indicate positive or negative sentiment?.**

display dictionary

/variables overall to q9.

## Result

If we quickly inspect these tables, we see two important things:

- for all statemtents, higher values indicate
**stronger agreement**; - all
**statements are positive**(“like” rather than “dislike”): more agreement indicates more positive sentiment.

Taking these findings together, we expect positive (rather than negative) correlations among all these variables. We'll see in a minute that our data confirm this.

## Data Check 2 - Distributions

Our previous table *suggests* that all variables hold values 1 through 11 and 11 (“No answer”) has already been set as a user missing value. Now let's see if the distributions for these variables make sense by running some histograms over them.

***Check distributions.**

frequencies overall to q9

/format notable

/histogram.

## Data Check 3 - Missing Values

First and foremost, the distributions of all variables show values 1 through 10 and they **look plausible**. However, we have 464 cases in total but our histograms show slightly lower sample sizes. This is due to missing values. To get a quick idea to what extent values are missing, we'll run a quick DESCRIPTIVES table over them.

***1. Show only variable labels in output.**

set tvars labels.

***2. Check for missings and listwise valid n.**

descriptives overall to q9.

## Result

For now, we mostly look at N, the number of valid values for each variable. We see two important things:

- The
**lowest N is 430**(“My workspace is good”) out of 464 cases; roughly 7% of the values are missing. - Only 297 cases have
**zero missing values**on all variables in this table.

## Correlations

We'll now inspect the correlations over our variables as shown below.

In the next dialog, we select all relevant variables and leave everything else as-is. We then click

, resulting in the syntax below.***Inspect correlations.**

CORRELATIONS

/VARIABLES=overall q1 q2 q3 q4 q5 q6 q7 q8 q9

/PRINT=TWOTAIL NOSIG

/MISSING=PAIRWISE.

Importantly, note the last line -`/MISSING=PAIRWISE.`

- here.

## Result

Note that all correlations are positive -like we expected. Most correlations -even small ones- are **statistically significant** with significance levels close to 0.000. This means there's a zero probability of finding this sample correlation if the population correlation is zero.

Second, each correlation has been calculated on all cases with valid values on the 2 variables involved, which is why each correlation has a different N. This is known as **pairwise exclusion of missing values**, the default for CORRELATIONS.

The alternative, **listwise exclusion of missing values**, would only use our 297 cases that don't have missing values on *any* of the variables involved. Like so, pairwise exclusion uses way more data values than listwise exclusion; with listwise exclusion we'd “lose” almost 36% or the data we collected.

## Multiple Linear Regression - Assumptions

Simply “regression” usually refers to (univariate) multiple linear regression analysis and it requires some assumptions:^{1,4}

- the prediction errors are
**independent**over cases; - the prediction errors follow a
**normal distribution**; - the prediction errors have a constant variance (
**homoscedasticity**); - all relations among variables are
**linear**and additive.

We usually check our assumptions before running an analysis. However, the regression assumptions are mostly **evaluated by inspecting some charts** that are created when running the analysis.^{3} So we first run our regression and then look for any violations of the aforementioned assumptions.

## Regression

Now that we're sure our data make perfect sense, we're ready for the actual regression analysis. We'll generate the syntax by following the screenshots below.

(We'll explain why we choose

when discussing our output.)
- Here we select some charts for evaluation the regression assumptions.

By default, SPSS uses only our 297 complete cases for regression. By choosing this option, our regression will use the correlation matrix we saw earlier and thus use more of our data.

## “Stepwise” - What Does That Mean?

When we select the stepwise method, SPSS will **include only “significant” predictors** in our regression model: although we selected 9 predictors, those that don't contribute uniquely to predicting job satisfaction will *not* enter our regression equation. In doing so, it iterates through the following steps:

- find the predictor that contributes most to predicting the outcome variable and add it to the regression model if its p-value is below a certain threshold (usually 0.05).
- inspect the p-values of all predictors in the model. Remove predictors from the model if their p-values are above a certain threshold (usually 0.10);
- repeat this process until 1) all “significant” predictors are in the model and 2) no “non significant” predictors are in the model.

## Regression Results - Coefficients Table

Our coefficients table tells us that SPSS performed 4 steps, adding one predictor in each. We usually report only the final model.

Our **unstandardized coefficients** and the constant allow us to predict job satisfaction. Precisely,
Y' = 3.233 + 0.232 * x1 + 0.157 * x2 + 0.102 * x3 + 0.083 * x4
where Y' is predicted job satisfaction, x1 is meaningfulness and so on. This means that respondents who score 1 point higher on meaningfulness will -on average- score 0.23 points higher on job satisfaction.

Importantly, all predictors contribute *positively* (rather than negatively) to job satisfaction. This makes sense because they are all positive work aspects.

If our predictors have different scales -not really the case here- we may compare their relative strenghts -the **beta coefficients**- by standardizing them. Like so, we see that meaningfulness (.460) contributes about twice as much as colleagues (.290) or support (.242).

All predictors are **highly statistically significant** (p = 0.000), which is not surprising considering our large sample size and the stepwise methode we used.

## Regression Results - Model Summary

Adding each predictor in our stepwise procedure results in a better predictive accuracy.

**R** is simply the Pearson correlation between the actual and predicted values for job satisfaction;

**R square** -the squared correlation- is the proportion of variance in job satisfaction accounted for by the predicted values;

We typically see that our regression equation performs better in the sample on which it's based than in our population. tries to estimate the predictive accuracy in our population and is slighly lower than R square.

We'll probably settle for -and report on- our final model; the coefficients look good it predicts job performance best.

## Regression Results - Residual Histogram

Remember that one of our regression assumptions is that the residuals (prediction errors) are normally distributed. Our histogram suggests that this more or less holds, although it's a little skewed to the left.

## Regression Results - Residual Plot

We also created a scatterplot with predicted values on the x-axis and residuals on the y-axis. This chart does not show violations of the independence, homoscedasticity and linearity assumptions but it's not very clear.

We mostly see a striking pattern of descending straight lines. This is because our dependent variable only holds values 1 through 10. Therefore, each predicted value and its residual always add up to 1, 2 and so on. Standardizing both variables may change the scales of our scatterplot but not its shape.

## Stepwise Regression - Reporting

There's no full consensus on how to report a stepwise regression analysis.^{5,7} As a basic guideline, include

- a table with
**descriptive statistics**; - the
**correlation matrix**of the dependents variable and all (candidate) predictors; - the model summary table with
**R square**and change in R square for each model; - the
**coefficients table**with at least the B and β coefficients and their p-values.

Regarding the correlations, we'd like to have statistically significant correlations flagged but **we don't need their sample sizes** or p-values. Since you can't prevent SPSS from including the latter, try SPSS Correlations in APA Format.

You can further edit the result fast in an OpenOffice or Excel spreadsheet by right clicking the table and selecting
.

I guess that's about it. I hope you found this tutorial helpful. Thanks for reading!

## References

- Stevens, J. (2002).
*Applied multivariate statistics for the social sciences.*Mahway, NJ: Lawrence Erlbaum Associates. - Agresti, A. & Franklin, C. (2014).
*Statistics. The Art & Science of Learning from Data.*Essex: Pearson Education Limited. - Hair, J.F., Black, W.C., Babin, B.J. et al (2006).
*Multivariate Data Analysis.*New Jersey: Pearson Prentice Hall. - Berry, W.D. (1993).
*Understanding Regression Assumptions.*Newbury Park, CA: Sage. - Field, A. (2013).
*Discovering Statistics with IBM SPSS*Newbury Park, CA: Sage. - Howell, D.C. (2002).
*Statistical Methods for Psychology*(5th ed.). Pacific Grove CA: Duxbury. - Nicol, A.M. & Pexman, P.M. (2010).
*Presenting Your Findings. A Practical Guide for Creating Tables*. Washington: APA.

## This tutorial has 5 comments

## By AMAR MISHRA on June 4th, 2017

lucid....thanks!

## By Ruben Geert van den Berg on May 29th, 2017

Hi Adriaan!

B coefficients can be lower or higher than beta coefficients

but they always have the exact same p-values.

The beta coefficients are

standardizedand the b coefficients areunstandardized.After standardizing all of your variables, the b and beta coefficients will be identical. Indeed, it's just a matter of scale as represented by the mean and standard deviation of the input variables.

Hope that helps!

## By Adriaan on May 29th, 2017

Thaks for your explanations are really good, but I would like to know more precisely the outcome of Coefficent. I understand that the Beta coefficient is associated with the standard deviations, I´m right? What is then the interpretation of this. Is it possible to have a higher b value with a poor Beta coefficient?

Thank you!

## By Ruben Geert van den Berg on March 3rd, 2017

Hi Jon, thanks for your feedback!

I'll try ALM a couple of times, see if it outperforms the stepwise regression substantively in practice.

I'm aware of the issues regarding stepwise regression -Stevens (2002) has a nice chart with theoretical versus actual F-values and warns against taking them too literally.

But from a pragmatic point of view, I think stepwise often strikes a good balance between still being comprehensible for non expert users and delivering solid results. At least we don't need to go into details why non significant predictors with "wrong signs" show up due to multicollinearity and they don't affect "healthy" beta coefficients.

Regarding pairwise deletion: I think this shouldn't be too much of a problem when the overall missingness percentage is low -as we confirmed for our data. "Losing" a large part of your sample through listwise deletion may also sacrifice its representativeness -a great downside that seems routinely neglected.

Perhaps multiple imputation would be the best option but I believe it requires an extra module that not all readers may have. So I like to keep things as simple as I can.

The CODEBOOK suggestion is great! I never used it but I surely will in future tutorials!

## By Jon Peck on March 2nd, 2017

Stepwise regression remains a very popular technique for situations where the correct model is not known from theory or prior research, but users should be aware of its drawbacks.

It is not guaranteed to find the best model, because it does not evaluate all possible models - which would be difficult if the number of candidate variable is very large. ALM (at the top of the Regression submenu) offers a best subsets search strategy without an excessive computational burden.

More importantly, stepwise results in exaggerated significance levels, so the results may be less significant than the reported p values suggest. And the fit measures are also exaggerated and are likely to be lower on new data. Using a holdout sample to test predictive accuracy is a good idea.

Stepwise does not consider interactions or nonlinearities (unless you compute interaction terms and included them in the candidate list. The SPSSINC CREATE DUMMIES extension command can create these for you.) Sometimes these are important.

Shrinkage methods such as ridge regression, lasso, and elastic net, while more difficult to understand, may give better results. These are available in CATREG if you have the Categories option.

On another aspect, I would be careful with the use of pairwise deletion of missing values. While this increases the number of usable cases, the statistical properties of regression done this way are unclear. The regressor crossproduct matrix might not even be positive definite,

Finally, rather than DISPLAY DICTIONARY for checking the variable coding, I suggest using CODEBOOK. It has the advantage of integrating appropriate summary statistics with the dictionary information.