- Cronbach’s Alpha - Quick Definition
- SPSS Cronbach’s Alpha Output
- Increase Cronbach’s Alpha by Removing Items
- Cronbach’s Alpha is Negative
- There are too Few Cases (N = 0) for the Analysis
- APA Reporting Cronbach’s Alpha

## Introduction

A psychology faculty wants to examine the reliability of a personality test. They therefore have a sample of N = 90 students fill it out. The data thus gathered are in big-5.sav, partly shown below.

As suggested by the variable names, our test attempts to measure the “big 5” personality traits. For other data files, a factor analysis is often used to find out which variables measure which subscales.

Anyway. Our main research question is: what are the reliabilities for these 5 subscales as indicated by Cronbach’s alpha? But first off: what's Cronbach’s alpha anyway?

## Cronbach’s Alpha - Quick Definition

Cronbach’s alpha is the extent to which the sum over 2(+)

variables measures a single underlying trait.
More precisely, Cronbach’s alpha is the proportion of variance of such a sum score that can be accounted for by a single trait. That is, it is the extent to which a sum score *reliably* measures something and (thus) the extent to which a set of items *consistently* measure “the same thing”.

Cronbach’s alpha is therefore known as a measure of **reliability** or **internal consistency**. The most common rules of thumb for it are that

- Cronbach’s alpha ≥
**0.80**is**good**and - Cronbach’s alpha ≈
**0.70**may or may not be just**acceptable**.

## SPSS Reliability Dialogs

In SPSS, we get Cronbach’s alpha from as shown below.

For analyzing the first subscale, agreeableness, we fill out the dialogs as shown below.

Clicking Paste results in the syntax below. Let's run it.

***CRONBACH AGREEABLENESS.**

RELIABILITY

/VARIABLES=agree01 agree02 agree03 agree04 agree05

/SCALE('Agreeableness') ALL

/MODEL=ALPHA

/STATISTICS=CORR

/SUMMARY=TOTAL.

## SPSS Cronbach’s Alpha Output I

For reliability, SPSS only offers listwise exclusion of missing values: all results are based only on N = 85 cases having zero missing values on our 5 analysis variables or “items”.

Cronbach’s alpha = 0.894. You can usually ignore Cronbach’s Alpha Based on Standardized Items: standardizing variables into z-scores prior to computing scale scores is rarely -if ever- done.

Finally, excluding a variable from a (sub)scale may increase Cronbach’s Alpha. That's not the case in this table: for each item, Cronbach’s Alpha if Item Deleted is lower than the α = 0.894 based on all 5 items.

We'll now run the exact same analysis for our second subscale, conscientiousness. Doing so results in the syntax below.

***CRONBACH CONSCIENTIOUSNESS.**

RELIABILITY

/VARIABLES=consc01 consc02 consc03 consc04 consc05

/SCALE('Conscientiousness') ALL

/MODEL=ALPHA

/STATISTICS=CORR

/SUMMARY=TOTAL.

## Increase Cronbach’s Alpha by Removing Items

For the conscientiousness subscale, Cronbach’s alpha = 0.658, which is pretty poor. However, note that Cronbach’s Alpha if Item Deleted = 0.726 for both consc02 and consc04.

Since removing either item should result in α ≈ 0.726, we're not sure which should be removed first. Two ways to find out are

- increasing the decimal places or (better)
- sorting the table by its last column.

As you probably saw, we already did both with the following OUTPUT MODIFY commands:

***Increase decimal places for Cronbach's alpha if item deleted.**

output modify

/select tables

/tablecells select = ['Cronbach''s Alpha if Item Deleted'] format = 'f10.8'.

***Sort item-total statistics by Cronbach's alpha if item deleted.**

output modify

/select tables

/table sort = collabel('Cronbach''s Alpha if Item Deleted').

It turns out that removing consc04 increases alpha slightly more than consc02. The preferred way for doing so is to simply copy-paste the previous RELIABILITY command, remove consc04 from it and rerun it.

***REMOVE CONSC04.**

RELIABILITY

/VARIABLES=consc01 consc02 consc03 consc05

/SCALE('Conscientiousness') ALL

/MODEL=ALPHA

/STATISTICS=CORR

/SUMMARY=TOTAL.

After doing so, Cronbach’s alpha = 0.724. It's not *exactly* the predicted 0.726 because removing consc04 increases the sample size to N = 84. Note that we can increase α even further to 0.814 by removing consc02 as well. The syntax below does just that.

***REMOVE CONSC02.**

RELIABILITY

/VARIABLES=consc01 consc03 consc05

/SCALE('Conscientiousness') ALL

/MODEL=ALPHA

/STATISTICS=CORR

/SUMMARY=TOTAL.

Note that Cronbach’s alpha = 0.814 if we compute our conscientiousness subscale as the sum or mean over consc01, consc03 and consc05. Since that's fine, we're done with this subscale.

Let's proceed with the next subscale: extraversion. We do so by running the exact same analysis on extra01 to extra05, which results in the syntax below.

***CRONBACH EXTRAVERSION.**

RELIABILITY

/VARIABLES=extra01 extra02 extra03 extra04 extra05

/SCALE('Extraversion') ALL

/MODEL=ALPHA

/STATISTICS=CORR

/SUMMARY=TOTAL.

## Cronbach’s Alpha is Negative

As shown below, Cronbach’s alpha = -0.663 for the extraversion subscale. This implies that some correlations among items are negative (second table, below).

All extraversion items are coded similarly: they have identical value labels so that's not the problem. The problem is that some items measure the opposite of the other items as shown below.

The solution is to simply reverse code such “negative items”: we RECODE these 2 items and adjust their value/variable labels with the syntax below.

***REVERSE CODE EXTRA01 AND EXTRA02.**

RECODE extra01 extra02 (1.0 = 5.0)(2.0 = 4.0)(3.0 = 3.0)(4.0 = 2.0)(5.0 = 1.0).

EXECUTE.

VALUE LABELS

/extra01 5.0 'Disagree strongly' 4.0 'Disagree a little' 3.0 'Neither agree nor disagree' 2.0 'Agree a little' 1.0 'Agree strongly' 6 'No answer'

/extra02 5.0 'Disagree strongly' 4.0 'Disagree a little' 3.0 'Neither agree nor disagree' 2.0 'Agree a little' 1.0 'Agree strongly' 6 'No answer'.

VARIABLE LABELS

extra01 'Tends to be quiet (R)'

extra02 'Is sometimes shy, inhibited (R)'.

Rerunning the exact same reliability analysis as previous now results in Cronbach’s alpha = 0.857 for the extraversion subscale.

So let's proceed with the neuroticism subscale. The syntax below runs our default reliability analysis on neur01 to neur05.

***CRONBACH NEUROTICISM.**

RELIABILITY

/VARIABLES=neur01 neur02 neur03 neur04 neur05

/SCALE('ALL VARIABLES') ALL

/MODEL=ALPHA

/STATISTICS=CORR

/SUMMARY=TOTAL.

## There are too Few Cases (N = 0) for the Analysis

Note that our last command doesn't result in any useful tables. We only get the warning shown below.

The 3 most likely **causes** for this problem are that

- one or more variables contains only missing values;
- an incorrect FILTER filters out
*all*cases in the data; - missing values are scattered over numerous analysis variables.

A very quick way to find out is running a minimal DESCRIPTIVES command as in descriptives neur01 to neur05. Upon doing so, we learn that each variable has N ≥ 67 but valid N (listwise) = 0.

So what we really want here, is to use pairwise exclusion of missing values. For some dumb reason, that's not included in SPSS. However, doing it manually isn't as hard as it seems.

## Cronbach’s Alpha with Pairwise Exclusion of Missing Values

We'll start off with the formula for Cronbach’s alpha, which is

$$Cronbach’s\;\alpha = \frac{k^2 \overline{S_{xy}}}{\Sigma S^2_x + 2 \Sigma S_{xy}}$$

where

- \(k\) denotes the number of items;
- \(S_{xy}\) denotes the covariance between each pair of different items;
- \(S^2_x\) denotes the sample variance for each item.

Note that a pairwise covariance matrix contains all statistics used by this formula. It is easily obtained via the regression syntax below:

***COMPUTE COVARIANCE MATRIX FOR NEUR01 THROUGH NEUR05.**

regression

/missing

**pairwise**

/dependent neur01

/method enter neur02 to neur05

/descriptives n

**cov**.

Next, we copy the result into this Googlesheet. Finally, a handful of very simple formulas tell us that α = 0.889.

Now, which sample size should we report for this subscale? I propose you follow the conventions for pairwise regression here and
report the smallest *pairwise* N
which results in N = 44 for this analysis. Again, note that the formula for finding this minimum over a block of cells is utterly simple.

## APA Reporting Cronbach’s Alpha

The table below shows how to report Cronbach’s alpha in APA style for all subscales.

This table contains the actual results from big-5.sav so you can verify that your analyses correspond to mine. The easiest way to create this table is to manually copy-paste your final results into Excel. This makes it easy to adjust things such as decimal places and styling.

Thanks for reading.

## THIS TUTORIAL HAS 19 COMMENTS:

## By Ruben Geert van den Berg on December 10th, 2023

Hi Tara!

First off, KR20 is identical to Cronbach. It is a simplified version of the formula (which was a time saver when people still calculated stuff without computers, long ago).

So if you have SPSS compute Cronbach's A via RELIABILITY, then it always uses listwise exclusion. In this case, take a good look at the CASE PROCESSING SUMMARY as it tells you the N that was used. Indeed, a super high alpha could result from a low N or maybe the correlations among variables are high -check these too! Also, you have quite a lot of variables -this also tends to yield (very) high alphas.

For pairwise exclusion, you can first compute a pairwise covariance matrix via REGRESSION. Next up, manually compute Cronbach on this matrix with a simple formula in Excel or Googlesheets. Report the lowest pairwise N as the N for Cronbachs alpha. This N is equal to DF(total) + 1 in the ANOVA table that comes with REGRESSION.

Does that answer your question?

Best,

Ruben

P.s. for questions like these, please throw in a comment under the tutorial on the website. Like so, other people can also read up on questions and answers.

## By T on December 13th, 2023

Dear Sir,

I am following your most excellent "Cronbach’s Alpha with Pairwise Exclusion of Missing Values" instructions. Thank you so much for making this information available.

My particular dataset has a much higher number of possible questions (21) than the number of questions actually shown to participants (6) (in other words, 15 missing values per participant). When I follow the instructions, I get "minimal N = 0".

I am trying to understand what that means from a practical standpoint. Is a min N of zero an acceptable number or would it invalidate or cause alpha to be questionable?

T

## By Ruben Geert van den Berg on December 14th, 2023

Hi Tara!

I'm not sure where you see the minimal N = 0 warning but with such a high percentage of missingness, alpha is indeed questionable.

This goes for all analyses that are based on the entire correlation/covariance matrix such as multiple linear regression and factor analysis.

In any case: if you run a standard (pairwise!) correlation matrix on your variables, you'll get the sample sizes for all variable pairs. If the lowest of these is zero, then that pair of variables can't be used so you need to remove at least one.

Also, you should report the minimal pairwise N as the sample size for the entire alpha analysis so if that's really low, then alpha is based on a small N and thus questionable.

Again, dropping variables may or may not reduce this problem. If it doesn't, then your data are simply unsuitable for this type of analysis.

Does that make any sense?

Best,

SPSS tutorials

## By T on December 14th, 2023

Thank you Ruben.

I received your reply. Even after removing variables where the sample size for at least one of the variable pairs is 0, I still get pairs where the sample size is 1.

I think most likely this is not the appropriate approach for my data set, unfortunately.

Thank you so much for your help.