Dit document bevat een uitwerking van hoorcolleges 1 t/m 5 en seminars 1 en 2. Deze hoorcolleges zijn stof voor het (midterm)tentamen van het General Part van het vak ARMS. Ik heb vrijwel alles meegeschreven, dus deze uitwerkingen kunnen goed gebruikt worden om te leren voor het tentamen. Aangezien...
ARMS (General Part)
HC1 | Irene Klugkist Multiple Linear Regression
Galton noticed that the number of firstborns among eminent scientists was remarkably large:
researchers observed a significant positive relation.
Critically review the way studies were performed!
- Representative sample?
- Reliable measures of variables?
- Correct analyses and correct interpretation of results
Critically consider alternative explanations for the statistical association!
- Association is not causation (the fact that things are related doesn’t mean that one causes
the other)
- Does effect remain when additional variables are included? This can be investigated with
multiple regression
Simple linear regression involves 1 outcome (Y) and 1 predictor (X)
- Outcome = DV = dependent variable = IQ
- Predictor = IV = independent variable = birth order
Multiple linear regression involves 1 outcome and multiple predictors
A good statistical model = when the model is a good way to describe the data and if the predictor is
useful for predicting your outcome. The relevance of a predictor:
1. The amount of variance explained (R2) = the sizes of the residuals
Can the predictor explain the IQ? If the line describes the dots very well than the model fits/explains
the data good = a larger R2 = scores of people are close to the line = a lot of variation is explained by
the model.
, 2. The slope of the regression line (B1)
How important is my predictor for predicting the outcome? The strength of the relation between X
and Y (=B1).
MLR examines a model where multiple predictors are included to check their unique linear effect on
Y. Need to know: the model, types of variables, MRL and hierarchical MLR (hypotheses, output,
model fit, regression coefficients), exploratory MLR (stepwise) vs confirmatory MLR (forced entry),
model assumptions important to MLR (see Grasple).
The model
Observed outcome is prediction based on the model and some error in prediction.
(1) observed scores (2) model (3) predicted scores. The predicted part is called the statistical model.
i = those variables where people differ from each other (personal error)
b = model parameters (so the terms without i)
^ = prediction
The relation between x1 and y, if I summarize over the whole sample, is b1. That is how x1 en y are
related if I take everything together, if I look at a scatterplot.
MLR-model is sometimes called an additive linear model = two predictors, for each of them (X1 and
X2) assuming that there is a linear relation with the outcome variable (Y) and that these effects add
up. For now, we look at additive effects, later also interaction-effects will be discussed.
, Types of variables
What types of variables can be included in MLR? That is a model assumption: what variables are
allowed in an MLR?
For choice of analysis we usually distinguish: does it create a group?
- “Nominal + Ordinal” aka categorical or qualitive
- “Interval + Ratio” aka continuous or quantitative or numerical
MLR requires continuous outcome and continuous predictors.
But! Categorical predictors can be included as dummy variables.
Dummy variable has only values 0 and 1 (not 1 and 2!). Example:
B1 represents the difference between males and females, between the averages. Normal we call B 1 a
slope of a linear line. That is weird with dummy variables. But it does note the difference in the
prediction between two groups. That has an interpretation that makes sense to us.
Gender is easy because it had two categories. With more categories (like color) the coding is like this.
Yellow = the reference group. In this example the intercept (B0) is the average on Y for the reference
group, because these people have zeroes on all dummies so they disappear. B1 is the difference in
the prediction between the reference group and group red.
MLR and hierarchical MLR
Example: What makes old people happy?
Research question 1: Can life satisfaction (y) be predicted from age (X1) and years of education (X2)?
Research question 2: Are social network factors (as measured by child support (X3) and spouse
support (X4)) improving the prediction of life satisfaction, if the effects of age and years of education
are already accounted for? + Is the addition useful? Does it improve the model significantly and
relevantly? = hierarchical MRL
Hypotheses Z.O.Z.:
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