Endogeniety

Endogeniety simply means that a parameter or variable is correlated with the error term. There are many reasons why this would happen:

1. Measurement error
• This happens when we do not have an accurate measure of the independent variables.
2. Omitted variables
• This happens when the model does not include all the variables it should, and thus we have an uncontrolled variable.
3. Simultaneity
• This happens when two variables are each affecting the other.

To address endogeniety, there are a few options:

• Use instrumental variables address omitted variables
• Heckman correction models address the sampling bias and unobservable variables
• If the data is not a panel, then propensity score matching could help a small sample
• Run a 2SLS or 3SLS

(Adapted from course notes)

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Fixed vs. Random Effects

The best way to think about the difference between random and fixed effects is with this picture.

Fixed effects can be thought of as the relationship between predictor and outcome within an entity. In addition:

• Assumes something about entity may bias predictor/outcome so need to control for it
• Removes effects of observed or unobserved time-invariant characteristics from predictor variables
• It helps with omitted variables bias
• Creates separate regressions for each entity and averages effects across entities

Random effects, on the other hand, vary across entities

• Assumes random and uncorrelated with IVs
• Can include time-invariant variables
• Assumes entity’s error term is not correlated with predictors which allows time-invariant variables can be explanatory variables

Some examples include:

• Time-varying observables – age, years of experience
• Time-invariant observables – degree, gender
• Time-invariant unobservables – ability, IQ
• Omitted variables are time invariant

(Adapted from course notes)

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Structural Equation Modeling

When you think that there are unobserved or latent variables, a potential technique is Structural Equation Modeling (SEM).

Among other advantages, SEM:

• Can control for random errors
• Can model measurement error so the model is more precise
• Can test elaborate models

In general, the SEM starts by looking at all of the relationships in your study. This is known as the perfectly saturated model. All other models are compared to this one. The chi square here should be zero.

From there, it is possible to fix some relationships based on theory. This model is aligning variables to constructs and it is similar to a factor analysis. This model is referred to as the measurement model. If this model is very good, then the chi square will be insignificant. Furthermore, we do not want to have a significant difference between the estimated covariance matrix of the measurement and saturated models. The measurement model is used to asses convergent/discriminant validity. However, this model does not say anything about causality.

It is then possible to specify a causal model using theory. In this case, we also want a low chi square statistic. That would suggest that there is no difference in the estimated covariance matrix from the theoretical model and of the observed.

Last is a further constrained model. This aims to get the most parsimonious model. In this case, some relationships are set to zero.  This model should have a larger chi square (bad news), but it is more parsimonious (good news). The goal is to determine if the change in the chi square between the theoretical model and this parsimonious model is significant. If the change is not significant, we should choose the parsimonious model.

(Adapted from course notes)

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Survival Analysis

Survival analysis is used when we need to choose a point in time to measure survival, success, failure, death, etc..

Two important concepts relate to the time that we observe the data. This is refered to as censoring and it comes in two varieties: left and right.

Left censoring happens when we do not know how many organizations, people, etc have failed before we began sampling. That means that we only see the people that have survived up to that point, and how long have they survived.

Right censoring happens when we do not know it the people survived and for how long beyond conclusion of the study.

Some techniques to look at survival analysis include:

• Hazard rate. This is the rate of not survivivngt to the midpoint of a specified time interval
• Cox Haphazard regression. This examines which IVs influence if failure occurred at a particular time. It is a proportional model, so time is not considered.
• A generic hazard model does not use proportion. So this model does think about how failure happens as a function of time.

(Adapted from course notes)

(Flashcards and other resources here)

Categorical DVs

When the DVs are categorical variables, different analyses should be used.

The most common (and the only one discussed in class) is the case of a binary outcome variable. If this is the case either Logit or Probit should be used. Logistic regression estimates the probability of the outcome variable having a certain value (as opposed tot he value itself).

When there are more than two outcome variables, we need multiple logistic regressions solved simultaneously.

Moderators and Mediators

A moderator is a qualitative or quantitative variable that affects the direction and/or strength of the relation between an IV (or predictor) and a dependent or criterion variable.

Within a correlational analysis framework, a moderator is a third variable that affects the zero-order correlation between two other variable. In the more familiar ANOVA terms, a basic moderator effect can be represented as an interaction between a focal IV and a factor that specifies the appropriate conditions for its operation. In other words, an observed relationship may be different at different levels of a third variable. Moderation refers to the situation where the direction and intensity of an effect of a predictor on a criterion depends on the levels or settings of a third variable.  In essence, moderators attenuate or exacerbate the effect.

To test:

1. Variables entered into the regression equation in a stepwise and hierarchical fashion.
2. Control variables (if any were collected) are entered first into the equation.
3. In order to derive main effects of X on Y, regress X onto Y for this step
4. Add the interaction terms (X x M) to the analysis.  If there is a change in total variance explained from step 3 to step 4 this suggests total moderational impact, and r2 values for each interaction term shows the impact of the moderator for each relationship.

A variable is said to function as a mediator to the extent that it accounts for the relation between the predictor and the criterion (X and Y). Mediators explain how external physical events take on internal psychological significance. Whereas moderator variables specify when certain effects will hold, mediators speak to how or why such effects occur. 

For a variable to be considered a mediator, it must pass three tests:

1. X correlates with Y
2. X correlates with M
3. M significant impact on Y when X controlled for
• Effect of X on Y when M controlled for is 0 for full mediation

This is mostly been adapted from Baron and Kenny, 1986

(Adapted from course notes)

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Analysis of Variance and Sundry

Here I will write very briefly about some analysis.

T-Test
This test compares the means to two groups. The goal is to determine if they are statistically different from one another.

Analysis of Variance
These include the MANOVA, ANOVA and MANCOVA and ANCOVA.

The MANOVA test compares the multivariate means of multiple groups. If this test is significant, then it is possible to do an ANOVA test for each DV.

In particular, the MANOVA:

• Has more than one DV
• Uses an omnibus F-test
• The IV is categorical
• Identifies significant DVs
• Assumes normality, linearity

ANOVA:

• Can be used for each DV after MANOVA F-test is significant.

The MANCOVA and ANCOVA test includes covariates. The covariates are used when you want to control for this variables in the analysis of variance. 

Regressions

The purpose of a regression is to estimate the relationship between the independent (predictor/explanatory) variables and the dependent (response./outcome) variable. Assumptions of regressions include:

• No specification error (no omitted variables)
• No measurement error
• There is no multicollinearity (variables are independent)
• Errors are independent (no omitted variables)
• Errors are normally distributed
• Normality

Hierarchical Regression

Are a type of regression models. This analysis builds successive linear regression models, each time adding more predictors. Here, the order of input matters. Generally, the oder is:

• Controls
• Main
• Interactions and higher order

(Adapted from course notes)

(Flashcards and other resources here)

Factor Analysis

The first steps to take with new data were already discussed in the previous post.

Factor analysis comes in two varieties: Exploratory and Confirmatory:

Exploratory factor analysis is used when you don't have a clear idea of what items might belong together. There are many ways to see what items belong together. The principal component analysis is most commonly used. Items that hang together must also have face validity.

Confirmatory factor analysis is more complicated. This analysis tests that the hypothesized items belong to particular factors.

(Adapted from course notes)

(Flashcards and other resources here)