Latent regression models

Where does the probability come from? Why isn’t the model deterministic. Each item tests something unique – We are interested in the average of what the items assess Stochastic subject argument Random sampling of subjects

Two different models Random sampling Stochastic subject

A Random Effects (Sampling) Model -- 1 Part 1: A Model for the population equivalently

A Random Effects Model -- 2 Part 2: A Model for the item response mechanism Example: SLM (but can be any)

A Random Effects Model -- 3 Part 3: Putting them together The unknowns are: is not an unknown, it is variable of integration What is analysed is item response, what is estimated is item parameters and population parameters

Why do this? Solves some theoretical estimation problems – For non-Rasch models Provides better estimates of population characteristics

Problems with point estimates

Problem with discreteness For a 6-item test, there are only 7 possible ability estimates to assign to people, those getting a score of 0,1,2,3,4,5,6. (raw score is sufficient statistic for ability) Suppose we want to know where the 25th percentile point is. That is, 25% of the population are below this point. We need extrapolation.

The Resulting JML Ability Distribution Score 0 Score 1 Score 2 Score 3 Score 4 Score 5 Score 6 Proficiency on Logit Scale

Distribution for a six item test Score 0 Score 1 Score 2 Score 3 Score 4 Score 5 Score 6 Proficiency on Logit Scale

Traditional approach is a two-step analyses First estimate abilities Then compute population estimates such as mean, variance, percentiles using leads to biased results due to measurement error. In the case of the population variance, we can correct the bias (disattenuate) by multiplying by the reliability. But in other cases, it is less obvious how to correct for the bias caused by measurement error.

Distribution of Estimates is Discrete One ability for each raw score Ability estimates have a discrete distribution We imagine (and the model’s premise) is a continuous distribution The distribution of ability estimates is distorted by measurement error

Solutions Direct estimation of population parameters (directly via item responses, and not through the estimated abilities) Complicated analyses that take into account the error – Not always possible

MML: How it works — 1 Item Response Model for item i: Population Model (discrete)

MML: How it works — 2

The Implications — 1 If, then contains parameters – Note that no ability parameters are involved, only population parameters. Use maximum likelihood estimation method to estimate the item difficulty parameters and population parameters. Thus, we directly estimate population parameters through the item responses

Bayes Theorem

The Idea of Posterior Distribution If a student’s item response pattern is x then the posterior distribution is given by

The Idea of Posterior Distribution Instead of obtaining a point estimate for ability, there is now a (posterior) probability distribution incorporates measurement error for the uncertainty in the estimate.

The Resulting JML Ability Distribution Score 0 Score 1 Score 2 Score 3 Score 4 Score 5 Score 6 Proficiency on Logit Scale

Resulting MML Posterior Distributions Score 0 Score 1 Score 2 Score 3 Score 4 Score 5 Score 6 Proficiency on Logit Scale

MML EAP Estimates – an aside Score 0 Score 1 Score 2 Score 3 Score 4 Score 5 Score 6 Proficiency on Logit Scale

MML EAP Estimates – an aside Biased at the individual level Discrete scale, bias & measurement error leads to bias in population parameter estimates Requires assumptions about the distribution of proficiency in the population

Distribution for a six item test Score 0 Score 1 Score 2 Score 3 Score 4 Score 5 Score 6 Proficiency on Logit Scale

Estimating proportions below a point based up posterior distributions

More General Form of the Model Item response model Population model

The underlying population distribution is a mixture of two normal distributions, with different means ( and ). Population Not Normal E.g., sample consists of grades 5 and 8.

where x=0 if a student is in group 1 and x=1 if a student is in group 2. In this case, we estimate, and. Note that is the difference between the means of the two distributions. That is, group 1 has mean (as x=0), and group 2 has mean (as x=1). Latent Regression - 1

Latent Regression - 2 We call “x” a “regressor”, or a “conditioning variable”, or a “background variable”. We can generalise to include many conditioning variables.