Prediction

Question 1

prediction

Answer 1

using scores or one variable (X) to predict scores on another variable (Y), based on known correlation between them rXY

• A method for predicting Y from X, using information about XY’s relationship

Question 2

Note on Prediction

Answer 2

First need a sample in which both pieces of information (X and Y) are available, have an established correlation

Then, use this information in a new sample where only X scores are available, in order to predict the Y scores

Question 3

Features of Prediction

Answer 3

• often temporal symmetry: X is measured at an earlier occasion than Y
• Y is referred to as a dependent or criterion variable
• X is referred to as an independent or predictor variable

Question 4

Simple VS Multiple Regression

Answer 4

• simple regression: one X and one Y
• multiple regression: more than one X

there is a focus on simple regression in this course

Question 5

Regression Equation

Answer 5

also called a prediction equation

y = a+bX (like high school)

• equation describes a straight-line that best fits the data points in 2 dimensional (X-Y) space

Question 6

Mapping Correlation to Prediction

Answer 6

y’ or y^ = predicted score on y
y’ = My

Step 1: Convert Mx (mean scores of X) to Zx, Zx = X-M / Sx

Step 2: equation in z-scores, Zy’ = rZx

Step 3: Zy’ to Y’ (predicted __), Y’ = My + Zy’Sy

Question 7

If r =+1.0 then Zy’ = Zx, a thought experiment

Answer 7

would never typically get r= +1.0

• what is being predicted would equal what has already occurred (possibly?)
• a perfect relationship between the two variables

Question 8

If r is between 0 and 1,

Answer 8

X provides some information to help us predict Y, but the relationship is not perfect (other factors and noise involved)

Question 9

Note 2 on Prediction

Answer 9

The amount by which our prediction differs from 0 will depend on the strength of r

Question 10

The optimal prediction is given by:

Answer 10

Zy’ = rZx (prediction equation in terms of z-scores)

two extremes = r=0, r=1

• in between is where the prediction of Y based on X lies

Question 11

The prediction equation on a graph

Answer 11

• regression line passes through origin and has a slope of r
• prediction equation describes the line of best fit through the scatter plot of Zy against Zx

Question 12

Building predictions from raw scores

Answer 12

Y’ = a+bX

where b = r sy/sx

where a = My - bMx

Question 13

Jamovi for Linear Regression

Answer 13

“estimate” refers to raw score regression, intercept = a, x = b

• jamovi will produce standardised estimates, which refer to the z-score equation, if you select standardised estimate under model coefficients

Question 14

r^2

Answer 14

r^2 = proportion of variance accounted for

how well we are making predictions, to know if we should keep using this model or not

y - My = y’-My + y-y’

Deviation = prediction + error (residual)

y scores from the mean = predicted scores from the mean + differences in predicted scores and real scores

Question 15

r^2 in equations

Answer 15

y - My = y’ - My + y - y’

Σ(y-My)^2 = Σ(y’-My)^2 + Σ(y-y’)^2

SS(total) = SS(regression) + SS(residual)

Question 16

Finding scores due to prediction (r^2)

Answer 16

r^2 = Σ(y’-My)^2/Σ(Y-My)^2

= ss (regression)/ ss(total)

proportion of variability in y scores associated with changes in X (i.e. due to prediction)

Question 17

Finding scores not due to prediction

Answer 17

1-r^2 = Σ(y-y’)^2/ Σ(y-My)^2

= ss(residual)/ss(total)

proportion of variability in Y scores not associated with changes in X (i.e. not due to prediction)

regression line is defined so that ss (residual) is minimised (based on least squares criterion)

no other straight line will generate a smaller ss (residual) than y’ = a+bX

Question 18

when error in prediction when r is large

Answer 18

• y values will cluster around y’ values
• a larger proportion of Sy^2 is accounted for by prediction

Question 19

Error in prediction when r is small

Answer 19

• y values will vary more widely around y’ values
• smaller proportion of Sy^2 is accounted for by prediction

Question 20

Note on r^2

Answer 20

r and r^2 convey information about how well we can predict scores on Y for the sample as a whole

• what about how well we are predicting individual subjects? first look at assumptions being made when undergoing a prediction analyses

Question 21

Assumptions for Linear Regression

Answer 21

in population, X and Y form a bivariate normal distribution

• both samples come from a normal distribution

For each X there is a normal distribution of Y scores

• y’ is what we expect y to be on average, under the circumstance of X
• we won’t get perfect y’ every time, but it will be normal, the highest likelihood is that y’ is closest to Y
• mean of that distribution is what y is represented as

Linearity

• X and Y is linearly related

Homoscedasticity

• variance of distribution of y scores for each X score is the same
• each y distributions should have the same deviation

Question 22

When assumptions of linear regression are met:

Answer 22

can use prediction to estimate:

• percentage of cases that are a certain distance from their predicted value
• probability of a score being a certain distance from its predicted value

Question 23

Standard Error of Estimate

Answer 23

SD of distribution of observed scores around corresponding predicted score

• measures predictive error
• under assumption of bivariate normality Syx is the standard deviation of normal distribution of y scores, for any value of X

equation:

Syx = √ Σ(y-y’)^2/n-2

alternative, (when large sample sizes)

Syx = Sy √ (n-1)(1-r^2) / n-2