IndexesAnthony Sealey

University of Toronto

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• Recall that, in the very first week of the course, we discussed the measurement of a concept.

• One of the key points we discussed was the idea that in some instances measures could be constructed using more than one indicator of a given concept.

e.g. Creating a Measure of ‘Social

Progressivism’ Social Progressivis

m

Outlooks on Gay Rights

Outlooks on Prostitution

Outlooks on Abortion

• When we use multiple indicators in order to measure a concept, we call the resulting concept an ‘index’. We can refer to the underlying dimension that we are attempting to measure as a ‘latent variable’.

• So in the previous example, we might combine the indicators of outlooks on gay rights, outlooks on abortion, and personal feelings of religiosity into an index that we use to measure the latent variable ‘social progressivism’.

• The major steps involved in the construction of indexes are:

1) Find a set of indicators that you think will be closely related to the underlying latent variable that you are interested in measuring.

2) Recode the indicators so that they have the same direction and range.

3) Test the indicators to determine how well they fit together.

4) Once you’ve identified a set of indicators that are a good fit for the underlying latent concept, combine them into a single index.

Step One: Finding Indicators

• While the construction of indexes is really quite simple, it often involves quite a bit of work. Much of this work often involves combing through the data set that you are interested in using in order to find suitable indicators of the key concepts you are looking for.

Step Two: Recoding Indicators

• Once you have identified indicators of the latent concept that you think will fit well together, the next step is to recode them in order to allow for greater comparability. To do this you need to consider both direction and range.

• When recoding in order to ensure that all of your indicators have the same direction, conceive of your latent variable in terms of one end of the spectrum of views that you intend to consider.

e.g. ‘Social Progressivism’

• Then recode each of your variables so that the higher values of your indicators represent this end of the spectrum of views that you are intending to measure.

• When recoding in order to ensure that all of your indicators have the same range, make sure that each of your indicators have the same minimum (0) and maximum (1), and that intermediary values are equally spaced in between.

• Then recode each of your variables so that the higher values of your indicators represent this end of the spectrum of views that you are intending to measure.

Step Three: Fitting Indicators

• Once you have recoded your indicators so that they each have the same direction and range, the next step is to test the indicators to determine how well they fit together. • In order to do this we perform a ‘reliability’ analysis.

• The key component of a reliability analysis is the Cronbach’s alpha score. There are two such scores for any given indicator, an unstandardized and a standardized score. We want to focus on the standardized score.

• Different researchers have different perspectives about how high these scores should be in order to conclude that a given index is sufficiently reliable.

• For the purposes of this course we will use a standardized alpha score of 0.50 as our cut-off.

Step Four: Combining Indicators

• Once we have tested the indicators to ensure that their fit is good, the next step is to combine them into a single indicator.

• In order to do this we simply add them up and divide by the number of indicators.

A worked example …

Step One ______________________

Finding Indicators

• Let’s try using World Values Survey data

that’s been collected on citizens’ views towards homosexuality, prostitution and abortion as indicators to build a measure of outlooks on ‘social progressivism’.

Step Two ______________________

Recoding Indicators

• First let’s take a look at the first potential

indicator, views towards homosexuality. Recall that we need to consider whether or not we need to recode for either direction or range. Here’s the distribution of the variable:

minimum value

maximum value

• We can see that his variable ranges from

1 to 10. In this case, a response of 1 indicates that the respondent thinks homosexuality is ‘never justifiable’ and a response of 10 indicates that he or she thinks that homosexuality is ‘always justifiable’.

• Do we need to recode for direction?

• Do we need to recode for direction?

In this case we do not, because those who believe that homosexuality is always justifiable are more ‘socially progressive’, and so higher scores on this indicator suggest higher levels of what we are trying to measure.

• Do we need to recode for range?

• Do we need to recode for range?

In this case we do, because the variable ranges from 1 to 10, while we want our indicators to range from 0 to 1.

• How would we do this? Well, we want to

turn the 1s into 0s and 10s into 1s. If we first subtract 1 from each value, this turns our 1s into 0s. But now our 10s will be 9s. To turn these into 1s, all we need to do is to then divide by 9!

• In other words, we could use this simple

SPSS code to create our first indicator of social progressivism:

compute socprogin1 = (homosexuality-1)/9.

• But what if we were creating a measure

of ‘moral traditionalism’ instead of a measure of ‘social progressivism’?

• If this were the case we would need to

recode for direction, as those who believe that homosexuality is always justifiable are less ‘morally traditional’, and so higher scores on this indicator suggest lower levels of what we’re trying to measure.

• How would we do this? Well, we want to

turn the low scores into high scores and the high scores into low scores. In other words, we want to turn 1s into 10s and 10s into 1s.

• How do we turn a 10 into a 1? Start with 11, and subtract 10

from it.

• How do we turn a 1 into a 10? Start with 11, and subtract 1

from it.

• In other words, we could use this simple

SPSS code to recode our first indicator of moral traditionalism for direction:

compute mortradin1 = 11 – homosexuality.

• Once we’ve done this, our newly-created

indicator will also range from 1 to 10, so we’ll next have to recode it for range …

Step Three ______________________

Fitting Indicators

• In order to determine how well our

indicators fit together, we perform a reliability analysis.

• To do this for our three indicators of

social progressiveness ‘socprogin1’, ‘socprogin2’, and ‘socprogin3’, use the following SPSS code:

RELIABILITY /var=socprogin1, socprogin2, socprogin3 /SCALE(’socprogrel') All /summary=ALL.

• Once we do this, we want to look at two

key components of the output: (1) the standardized Cronbach’s alpha coefficient, and (2) the ‘alpha if item deleted’ score for each of the indicators.

• These three indicators combine for a

standardized alpha score of 0.728, which exceeds our 0.5 threshold. These indicators fit quite well together.

• These three indicators combine for a

standardized alpha score of 0.728, which exceeds our 0.5 threshold. These indicators fit quite well together. standardiz

ed alpha

• If one of our ‘alpha if item deleted’

scores is higher than this standardized alpha value, this suggests that our measure would be a better fit if we removed this indicator.

• Since each of these values is below 0.728,

this means that each of our indicators positively contributes to the fit of the measure.

• Since each of these values is below 0.728,

this means that each of our indicators positively contributes to the fit of the measure.

alpha if deleted for first

indicator

• Since each of these values is below 0.728,

this means that each of our indicators positively contributes to the fit of the measure.

alpha if deleted

for second indicator

• Since each of these values is below 0.728,

this means that each of our indicators positively contributes to the fit of the measure.

alpha if deleted for third indicator

Step Four ______________________

Combining Indicators

• The final step is to combine our

indicators to create a new measure. In order to do this, we want to simply sum up our indicators and then divide by the number of indicators:

• To do this for our three indicators of

social progressiveness ‘socprogin1’, ‘socprogin2’, and ‘socprogin3’, use the following SPSS code:

compute socprog = (socprogin1+socprogin2+socprogin3)/3.

• Once we’ve created our new measure of

social progressivism, take a look at it by running a frequency:

freq var = socproginx.