Learn About Hierarchical Linear Regression in SPSS With ...

Learn About Hierarchical Linear

Regression in SPSS With Data

From Prison Inmates

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This PDF has been generated from SAGE Research Methods Datasets.

Learn About Hierarchical Linear

Regression in SPSS With Data

From Prison Inmates

Student Guide

Introduction

This example dataset introduces hierarchical linear regression, which is a

statistical technique used to explore the relationship between one outcome or

“criterion” variable (i.e., suicide ideation) and several predictor variables (i.e.,

depression, hopelessness, thwarted belongingness, perceived burdensome).

This example describes the nature and purpose of a hierarchical linear regression,

discusses the underlying assumptions of the analysis, and explains how to

conduct and interpret the findings of this analysis. We then illustrate a hierarchical

linear regression using data from a study testing the leading hypothesis of the

interpersonal theory of suicide among adult male prisoners (see Mandracchia

& Smith, 2015). Specifically, we examine the ability of a combination of two

maladaptive interpersonal cognitions (i.e., thwarted belongingness and perceived

burdensomeness) to statistically predict suicide ideation, after statistically

controlling for depression and hopelessness. This analysis is useful as it could

provide support for this theory as a comprehensive explanation for suicide ideation

by expanding its applicability to this high-risk population (i.e., incarcerated

offenders).

What Is Hierarchical Linear Regression?

Assessing the degree to which variables are related typically involves researchers

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examining the strength of associations between those variables using simple

correlations, simple linear regressions, and multiple linear regressions.

Hierarchical linear regression is a special case of a multiple linear regression in

which additional variables are entered into the equation in subsequent “blocks” to

draw conclusions about how these added predictor variables change the model’s

ability to statistically predict the criterion variable. This is often used in order to

evaluate whether one or more variables statistically predict a criterion variable

while statistically “controlling for” one or more other variables. For example,

after statistically controlling for socioeconomic status and level of educational

attainment, does race and gender predict US worker’s salaries? This is also

often used to investigate the potential presence of a moderating relationship of

a variable. For example, does the interaction of age and alcohol consumption

predict students’ grade point average (GPA) above and beyond age and alcohol

separately? The mathematical equations below represent two “blocks” of a

hypothetical hierarchical linear regression in which the second equation includes

the original predictor variables from the first equation along with an added

predictor variable (Tabachnick & Fidell, 2013).

(1)

Y ′ = A + β1X1 + β2X2 + e

(2)

Y ′ = A + β1X1 + β2X2 + β3X3 + e

In this dataset, we will discuss how to perform a hierarchical linear regression. As

a review, the regression equation defines Y ′ as the predicted value of the criterion

variable. The values of Xk are the predictors. A is the value of the y-intercept when

all the predictor variables have values of 0. The values of β are the best fitting

coefficients (also known as regression/beta coefficients or beta weights) assigned

to each predictor during the regression, and e represents the residuals (i.e.,

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error). This approach was used with the example data to investigate whether the

interaction of thwarted belongingness and perceived burdensomeness statistically

predicts suicidal ideation after statistically controlling for depression and

hopelessness among adult male prisoners. In this case, Y ′ would be the

predicted value of suicide ideation scores. Depression scale scores (X1) and

hopelessness scale scores (X2) are entered in Block 1, thwarted belongingness

scale scores (X3) and perceived burdensomeness scale scores (X4) are entered

into Block 2, and the interaction term of thwarted belongingness scale scores

multiplied by perceived burdensomeness scale scores (X5) is entered into Block

3.

Model Improvement

In hierarchical linear regression, variables that have been previously established

in the literature as being related to the criterion variable are entered into the first

block, followed in subsequent blocks by the remaining variables in order of best

known to least known. The idea being that the least known variables are the new

variables to explore as predictors. When one or more new variables are entered

into a subsequent block of the hierarchy, this produces a new, expanded model

(i.e., multiple linear regression equation). Each model produces a best fit line,

just as in other forms of regression. A comparison of the best fit lines of every

model (i.e., for each block in the hierarchy) is needed to identify the model with the

best fit to the data. As new variables are entered into the levels of the hierarchy,

the test of whether or not the model significantly improves is the objective of the

analysis. In other words, the models are compared to see whether the added

new variable(s) significantly add(s) to the ability of the model to statistically predict

the criterion variable. The improvement in predictability is evaluated based on

the extent to which R2 (i.e., the variance accounted for in the criterion variable)

increases significantly between each model of the hierarchy (Field, 2013). To

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determine whether the change in R2 is statistically significant, the F-ratio equation

is used:

(3)

Fchange = [(N – κnew – 1)R2

change] / [κchange(1 – R2

new)]In this equation, N is the total sample size, κnew is the number of new variables

added in the new model, and κchange is the difference in the number of variables

between the two blocks being compared (Field, 2013). The change in the F-ratio

is examined to determine statistical significance and whether the new model fit is

significantly improved by adding the new variable(s).

Assumptions

The main assumptions of hierarchical linear regression are the same as for other

forms of regression analyses. This includes that multicollinearity does not exist

or is only present at very low levels (Tabachnick & Fidell, 2013); this can be

assessed using the Durbin–Watson test for independence of residuals (Field,

2013). Researchers should also check plots of standardized residuals to evaluate

for assumptions of normality, homoscedasticity, linearity, independence of errors,

and absence of outliers.

Illustrative Example: Interpersonal Beliefs and Suicide Ideation

This example examines the relationship between beliefs about interpersonal

needs and suicide ideation in a sample of adult male prison inmates. Specifically,

it evaluates the ability of two independent but related beliefs from the interpersonal

theory of suicide to statistically predict suicide ideation. The two beliefs include

a diminished sense of belonging with other people (i.e., thwarted belongingness)

and the sense of being a burden to other people (i.e., perceived

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burdensomeness). The interpersonal theory posits that the strongest desire to

die (i.e., suicide ideation) is created by the combined experience of thwarted

belongingness and perceived burdensomeness. Because depression and

hopelessness have historically been widely stated as causes of suicide ideation,

the interpersonal theory states that thwarted belongingness and perceived

burdensomeness promote suicide ideation above and beyond depressed mood

and general hopelessness.

The research question for this example is:

• Among adult male prisoners, does the combination of thwarted

belongingness and perceived burdensomeness statistically predict suicide

ideation after accounting for depression and hopelessness?

The null Hypothesis is:

When statistically controlling for both depression and hopelessness, the

interaction term for thwarted belongingness and perceived

burdensomeness will not be statistically significantly related to suicide

ideation among adult male prisoners.

The Data

Data were obtained from 399 adult male prisoners and were collected via the

following self-report surveys with Likert-type response options:

• Depression was operationally defined as the total score from the Center for

Epidemiological Studies–Depression Scale (CES-D; Radloff, 1977); it has a

possible range of 0–60.

• Hopelessness was operationally defined as the score from the

Hopelessness Scale of the Depression, Hopelessness, and Suicide

Screening Form (DHS-H; Mills & Kroner, 2004); it has a possible range of

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0–10.

• Thwarted belongingness was operationally defined as the score from the

thwarted belongingness scale of the Interpersonal Needs Questionnaire

(INQ-TB; Van Orden, Cukrowicz, Witte, & Jioner, 2012); it has a possible

range of 9–63.

• Perceived burdensomeness was operationally defined as the score from

the perceived burdensomeness scale of the Interpersonal Needs

Questionnaire (INQ-PB; Van Orden et al., 2012); it has a possible range of

6–42.

• The interaction term of thwarted belongingness and perceived

burdensomeness was created by multiplying the total scores of these two

variables together for each participant; it has a possible range of 54–2,646.

• Suicide ideation was operationally defined as the total score from the Beck

Scale for Suicide Ideation (BSI); it has a possible range of 0–42 (Beck &

Steer, 1993).

The below histograms provide a visual display of the data distributions for the

variables in the hierarchical linear regression. In these figures, the mean and

standard deviation for each of the variables are also present.

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Analyzing the Data

The hierarchical linear regression was conducted in IBM® SPSS® Statistics

software (SPSS). Suicide ideation scores were entered as the criterion variable.

Depression scores and hopelessness scores were entered into the first block,

thwarted belongingness scores and perceived burdensomeness scores were

added in the second block, and the interaction term (thwarted belongingness ×

perceived burdensomeness) was added in the third block. Table 1 provides the

results from this analysis:

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Table 1: Results From the Hierarchical Linear Regression Predicting

Suicide Ideation.

Predictor ΔF(p) ΔR2 b β t (p)

Block 1 72.325 (<.001) .271

CES-D .125 .274 5.185 (<.001)

DHS-H .606 .312 5.906 (<.001)

Block 2 19.568 (<.001) .067

CES-D .079 .174 3.264 (.001)

DHS-H .413 .213 3.890 (<.001)

INQ-PB .186 .312 6.102 (<.001)

INQ-TB .001 .003 .056 (.955)

Block 3 6.147 (.014) .010

CES-D .083 .183 3.438 (.001)

DHS-H .381 .197 3.587 (<.001)

INQ-PB .017 .029 0.230 (.818)

INQ-TB −.057 −.129 −1.794 (.074)

INQ-PB × INQ-TB .005 .379 2.479 (.014)

Note: Bold values are statistically significant.

Presentation and Additional Interpretation of the Results

In regard to the hypothesis, the results of the analysis could be presented as:

“A hierarchical linear regression analysis was conducted to test the hypothesis

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that the interaction term for thwarted belongingness and perceived

burdensomeness would be statistically significantly related to suicide ideation

when statistically controlling for both depression and hopelessness. Scores for

depression and hopelessness were entered into the first block of predictor

variables, scores for thwarted belongingness and perceived burdensomeness

were added into the second block, and the interaction term (i.e., scores for

thwarted belongingness multiplied by scores for perceived burdensomeness) was

added into the third block. As predicted, the interaction term was significantly

related to suicide ideation (t = 2.479, p = .014), above and beyond the statistically

significant relations between depression (t = 5.185, p < .001) and hopelessness (t

= 5.906, p < .001) to suicide ideation, respectively. Overall, the full model (i.e., the

third block) accounted for nearly 35% of the variance in suicide ideation scores

(i.e., total R2 = .348).”

The information from the hierarchical linear regression analysis presented in Table

1 indicates the amount of variance in suicide ideation scores accounted for each

of the three blocks. Because each block of the hierarchical regression is its

own model, the increased ability of each model (i.e., each successive block in

the hierarchy) to statistically predict the criterion variable is presented as the

change in R2 values (ΔR2). In Block 1, which contains only depression and

hopelessness as predictors, the model accounts for 27.1% of the variance in

suicide ideation. When thwarted belongingness and perceived burdensomeness

are added to the model in Block 2, together all four predictors account for 33.8%

of the variance in suicide ideation (i.e., an additional 6.7%; R2 = .067). Finally,

when the interaction term (thwarted belongingness × perceived burdensomeness)

is added to the model in Block 3, together all five predictors account for 34.8% of

the variance in suicide ideation (i.e., an additional 1%; R2 = .010). As noted by

the change in F statistic (ΔF) for each block, each block is statistically significant

in providing added predictive ability for suicide ideation scores (above and beyond

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the previous block).

Review

Using self-report data collected from a sample of adult male prisoners, the

interaction between the maladaptive interpersonal cognitions of thwarted

belongingness and perceived burdensomeness was examined in relation to

suicide ideation. Hierarchical linear regression analysis is a robust approach to

assess the theoretical considerations of the interpersonal suicide. The analysis

provides evidence that participants’ experiences of thwarted belongingness and

perceived interactively predict suicidal ideation after controlling for participants’

depression and hopelessness.

Dataset learning outcomes:

• The type of research questions hierarchical liner regression can answer.

• Considerations and limits of the analysis.

• How to conduct a hierarchical linear regression.

• How to interpret change in R-squared.

Your Turn

You can download the sample dataset along with the How-to Guide showing you

how to conduct a hierarchical linear regression in SPSS. The sample dataset

includes the same variables as the illustrative example along with a new variable

called CES-D × DHS that represents the interaction term of depression and

hopelessness. Following the steps outlined in the How-to Guide, try running a

hierarchical linear regression in SPSS with the control variables and predictor

variables switched. That is, use thwarted belongingness and perceived

burdensomeness as the control variables and depression, hopelessness, and

their interaction term as the predictors of suicidal ideation. For additional practice

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conducting hierarchical linear regression in SPSS, you could then try to reproduce

the results presented in the illustrative example (i.e., with depression and

hopelessness as “control” variables, etc.).

References

Beck, A. T., & Steer, R. A. (1993). Manual for the Beck scale for Suicide Ideation.

San Antonio: Psychological Corporation.

Field, A. P. (2013). Discovering statistics using IBM SPSS statistics: And sex and

drugs and rock ‘n’ roll (4th ed.). Los Angeles, CA: SAGE.

Harrell, F. (2018, April 10). Problems caused by categorizing continuous

variables. Retrieved from http://biostat.mc.vanderbilt.edu/wiki/Main/

CatContinuous

Mandracchia, J. T., & Smith, P. N. (2015). The interpersonal theory of suicide

applied to male prisoners. Suicide & Life-Threatening Behavior, 45(3), 293–301.

Retrieved from http://doi.org/10.1111/sltb.12132

Mertler, C. A., & Vannatta, R. A. (2013). Advanced and multivariate statistical

methods (5th ed.). Glendale, CA: Pyrczak.

Mills, J. F., & Kroner, D. G. (2004). A new instrument to screen for depression,

hopelessness, and suicide in incarcerated offenders. Psychological Services, 1,

83–91. doi:http://dx.doi.org/10.1037/1541-1559.1.1.83

Radloff, L. S. (1977). The CES-D Scale: A self-report depression scale for

research in the general population. Applied Psychological Measurement, 1,

385–401. doi:http://dx.doi.org/10.1177/014662167700100306

Tabachnick, B. G., & Fidell, L. S. (2013). Using multivariate statistics (6th ed.).

Boston, MA: Pearson Education.

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http://biostat.mc.vanderbilt.edu/wiki/Main/CatContinuous

http://biostat.mc.vanderbilt.edu/wiki/Main/CatContinuous

http://doi.org/10.1111/sltb.12132

http://dx.doi.org/10.1037/1541-1559.1.1.83

http://dx.doi.org/10.1177/014662167700100306

Van Orden, K. A., Cukrowicz, K. C., Witte, T. K., & Joiner, T. E. (2012).

Thwarted belongingness and perceived burdensomeness: Construct validity and

psychometric properties of the Interpersonal Needs Questionnaire. Psychological

Assessment, 24(1), 197–215. doi:http://dx.doi.org/10.1037/a0025358

Van Orden, K. A., Witte, T. K., Cukrowicz, K. C., Braithwaite, S. R., Selby,

E. A., & Joiner, T. E. (2010). The interpersonal theory of suicide. Psychological

Review, 117, 575–600. doi:http://dx.doi.org/10.1037/a0018697

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http://dx.doi.org/10.1037/a0025358

http://dx.doi.org/10.1037/a0018697

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