1. Operational definitions and latent variables As we discussed in our last class, many psychological variables of interest cannot be directly observed These latent variables can be quantified by (a) measuring the observable manifestations of these variables and (b) explicating the relationship between the latent and observable variable(s)

2. > 1One Linear Nonlinear Multiple non- linear indicators (Very Complex) Single non- linear relationship (Complex) Multiple linear indicators (Simple) Equivalence relation (Simplest) How many indicators? Mathematical Mapping

3. attraction heart beattalkingphone calls In our example, we were assuming that when someone is attracted to someone else (a latent variable), that person is more likely to have an increased heart rate, talk more, and make more phone calls (all observable variables).

4. attraction heart beat attraction talking attraction phone calls We assume that each observed variable has a linear relationship with the latent variable. Note, however, that each observed variable has a different metric (one is heart beats per minute, another is time spent talking). Thus, we need a different metric for the latent variable.

5. allow the lowest measured value to represent the lowest value of the latent variable allow the highest measured value to represent the highest value of the latent variable the line between these points maps the relationship between them Latent Observed

6. attraction heart beat attraction talking attraction phone calls After the relationship has been specified between the latent variable and each measured variable, the latent score estimates for each measured variable can be averaged to scale the person on the latent variable. estimate: 2estimate: 0estimate: 3 (2 + 0 + 3)/3 = 5/3 = 1.67 observed: 12observed: 10observed: 13

7. Multiple linear indicators Advantages –By using multiple indicators, the uniqueness of each one gets washed out by what is common to all of them Disadvantages –More complex to use –There is more than one way to scale the latent variable, thus, unless a scientist is very explicit, you might not know exactly what was done to obtain the measurements.

8. Multiple linear indicators: Caution On that last note, I should mention an important problem. When using multiple indicators, researchers typically sum or average the scores to scale people on the construct Example: (time spent talking + heart rate)/2 = attraction Person A: (2 + 80)/2 = 82/2 = 41 Person B: (3 + 120)/2 = 123/2 = 62

9. Multiple linear indicators: Caution Why may this be a problem? First, the resulting metric for the latent variable doesn’t make much sense. Person A: 2 minutes talking + 80 beats per minute = 41 minutes talking/beats per minute???

10. Multiple linear indicators: Caution Second, the variables may have different ranges. If this is true, then some indicators will count more than others.

11. Multiple linear indicators: Caution Variables with a large range will influence the latent score more than variable with a small range PersonHeart rate Time spent talking Average A80241 B80342 C120261 D120362 * Moving between lowest to highest scores matters more for one variable than the other * Heart rate has a greater range than time spent talking and, therefore, influences the total score more (i.e., the score on the latent variable)

12. Mapping the relationship by placing anchors at the highest and lowest values helps to minimize this problem Preview: Standardization and z-scores Latent Observed

13. Some more examples Let’s work through a detailed example in which we try to scale people on a latent psychological variable For fun, let’s try measuring stress: Some people feel more stressed than others Stress seems to be a continuous, interval-based variable What are some indicators of stress?

14. Some possible indicators of stress Hours of sleep Number of things that have to be done by Friday

15. Operationalizing our indicators We can operationally define these indicators as responses to simple questions: –“Compared to a good night, how many hours of sleep did you lose last night?” –“Please list all the things you have to accomplish before Friday— things that you can’t really put off.” Note that each of these questions will give us a quantitative answer. Each question is also explicit, so we can easily convey to other researchers how we measured these variables.

16. Latent: Stress Level Observed: Hours of Lost Sleep 6 4.2 2.4 -.6 -1.2 -3 Operationally defining the latent variable

17. Latent: Stress Level Observed: Things to do 15 12.6 10.2 7.8 5.4 3 Operationally defining the latent variable

18. Estimating latent scores PersonIndicator 1 (hours sleep) Latent score estimate 1 Indicator 2 (to do list) Latent score estimate 2 Averaged latent score Marc15413 Shamara5915109.5 c d e

19. Summary So, we find that Shamara has a higher stress level (9.5) than Marc (3). Recap of what we did –Determined the metric of the latent variable –Identified two indicators of the latent variable –Mapped the relationship between the latent variable and each observed variable –Using this mapping, estimated the latent scores for each person with each observed variable –Averaged the latent score estimates for each person