Predictions

Página 1

Prediction, IV & DV

Prediction

Relationship between an independent variable and a dependent variable.

Asymmetric where placement of variables matter.

Regression model.

Recall:

Association is symmetric where placement of variables does not matter.

Continuous variable - Correlation

Categorical variable - Contingency table, chi square, odds ratio

What are the similarities and differences between association and predictions?

Straight line functions

Predictions can be expressed as straight line functions.

y = m x + a

x is the independent variable

m is the constant for the slope

a is the intercept

y is the dependent variable

\(\widehat {y} \) denotes a y value that is on the line.

y denotes y value corresponding to a scatterpoint dot.

How do we get the best fit line for the scatterpoint dots?

By regressing the Y variable on the X variable (i.e. how much does the observed y value deviate from the predicted \(\widehat {y} \) value?.

The deviation in observed and predicted y values is called the residual value.

Note: The sum of all residual scores always = 0.

By using the least squares estimator where the min sum of \((y - \widehat {y})^2 \).

It is also known as the least squares regression function.

Linear regression model

1. Full simple regression equation (y)

2. Simple regression equation (\(\widehat {y} \))

b is the regression coefficient.

e = Y - \(\widehat {Y} \)

Both regression equations depend on the no of IVs.

Simple regression models contains 1 IV

Multiple regression models contain 2 or more IVs.

b in multiple regression equations are known as partial regression coefficients.

Each IV has their separate coefficient.

Página 2

Strength of prediction, R-squared, regression coefficients

Interpreting linear regression models

We can derive the following:

Strength of overall prediction

Strength of each individual IV

Note: Strong overall model does not imply that each IV is a good predictor.

Strong individual IV does not imply model is strong overall.

R-squared

\(R^2 \) statistic measures the overall strength of prediction for the model.

It measures the variability of scores of the DV in the regression model.

How is it derived?

3 types of sum of squared variations can be obtained.

Variation in observed scores Y

Variation in predicted scores \(\widehat {Y} \)

Variation in differences between Y and \(\widehat {Y} \)

Where 0 \(\leq\) \(R^2 \) \(\leq\) 1.

To take it a step further, we can determine the correlation strength between observed and predicted DV (Y) scores.

Multiple correlation coefficient

R = \(\sqrt{R^2}\) where 0 \(\leq\) \(R^2 \) \(\leq\) 1.

It is interpreted where higher observed scores for DV is associated with higher predicted scores for DV.

Strength of individual IV for simple and paritial

It can be assessed in 3 ways

Unstandardised regression coefficient

Standardised regression coefficient

Semipartial correlation (multiple regression)

Unstandardised simple regression coefficient

Expected change in raw scores of the DV with 1 unit change in the raw scores of the IV.

Interpreting the coefficient

Size

Direction

+ coefficient - Increase in IV results in an increase in DV

- coefficient - increase in IV results in a decrease in DV

Standardised simple regression coefficient

If all the raw scores were changed to standardised form, i.e. Z scores...

where \(\beta\) denotes standardised regression coefficient.

Note: Intercept is left out as it is always 0 in standardised form.

Definition:

Expected change in Z score units on the DV for 1 SD change on IV.

Interpretation of coefficient is the same as unstandardised.

Unstandardised partial regression coefficient

Definition: The expected change in the DV with 1 unit change in the focal IV given that all other IVs are held constant.

Likewise, a standard partial regression coefficient is defined as the expected change in the Z score of the DV with 1 SD change in the focal IV given that all other IVs are held constant.

Finally...

Regression intercept

Definition: Expected value of the DV when the scores on all IV (given more than 1 IV) is 0.

Note: Intercept = 0 in standardised regression equation.