psych 218 midterm 2 (regression)

thing youre trying to predict is called + what can be akin to IV’s

• criterion score (the observed score) 

• predictors

• Regression – the boring case 

• When the correlation between a predictor and a criterion is (positive or negative) 1.00, you can predict the criterion perfectly 

  • E.g. how much you have to pay in tax, based on the cost of the purchases = cost of purchase x tax rate = how much tax to pay 

Regression

• Predicting the criterion (Y) from one or more predictors (X1, X2, X3) 

• Simple regression → when there is only one predictor 

• Multiple regression → multiple predictors 

Line of best fit + how to find it

• line that minimizes the total error (deviation of each of the points from the line) 

• finding —> total the discrepancies and see between which graph has a smaller value (this is kinda by eyeballing)

• 

Least squares regression line

• A prediction line that minimizes the sum of the squared errors

• 

• Y = observed score, Y hat = predicted score

• Sigma = add all the errors together 

• For any linear relationship, there is only one line that will minimize ^

regression line equation

• example:

  

• Calculating (By) – the regression coefficient  (if you already have r and the standard deviations)

• If you already have the correlation coefficient ® and the standard deviations: 

  • Sx = standard deviation for X 

  • Sy = standard deviation for Y 

  • R = correlation between X and Y 

• Calculating (By) – the regression coefficient  (if you dont have r)

What does slope tell you

• For every 1 unit increase in X, the predicted value of Y changes by b amount 

  • E.g. b = +0.50 

    • If X = 1, revise your prediction for Y up by 0.50 

    • If X = 2, revise your prediction for Y up by 1.00 

  • E.g. b = -7.00 

    • If X = 1, revise your prediction of Y down by 7.00 

    • If X = -1, revise your prediction of Y up by 7.00 

Calculating (ay) intercept

• 

• Y bar = mean of Y 

• B = slope 

• X bar = mean for X

If the correlation between the criterion (Y) and the predictor (X) is zero, the regression line will be:

• a horizontal line 

• Limits of regression 

• Only appropriate to use regressions for: 

  • For linear relationships 

  • When the sample you used to calculate the regression line is representative of the sample you want to make predictions about 

  • Within the range of the original variables 

 standard error of estimate (SEE)

•  tells us how much error on average can we expect when we use the regression line → basically, how much are we off by? 

• smaller SE means better prediction (0 = perfect regression) 

SEE equation /formula

• (sum of squares = sum of deviations) = (x - mean of x)^2 then add all the values for each variable

How to phrase SEE

•  on average, the observed scores deviate from the predicted scores by x amount 

• e.g.

  

SE in relation to normal distribution?

• If SE= 2.93 (example number btw), we can expect about 68% of actual/observed Y scores to fall within 2.93 points of the prediction

  

• The standard error of estimate is computed over the whole range of data 

  • Thus, standard errors are only meaningful if the variability in Y is constant over values of X

Homoscedasticity

• If the spread is consistent across all the values of X. then you would say that you have Homoscedasticity and you can interpret SE as normal 

• 

This one ^ we do not have homoscedascitiy, and cannot interpret standard error.

Multiple regression

• In ^, we are using 2 or more predictors to predict a criterion (Y) 

• Goal: increase accuracy in predicting criterion

• 

R

• the maximum correlation between Y and the combination of predictors (X1 and X2 and so on) 

If you have 2 regressions and theyre both good at predicting scores — what do you choose

• go with the simpler one – helps with parsimony 

R² — Multiple coefficient of determination

• R^2 tells us the proportion of the variance in Y that is accounted for by the combination of predictors (X1 and X2) and so on) 

  • Lowest number R^2 can be: 0 (if you can explain none of the variability) 

  • Highest number R^2 can be: 1 or 1% (if you can explain all of the variability)

• if R² goes up, you can tell a regression has improved. also look at standard error, as it goes down ur regression improves

• formula:

  

• 44% of variability in happiness can be accounted for by our predictors income and optimism

can you add up r² to get total proportion of variability accounted for?

• You cant add up r^2 to get total proportion of variability accounted for because youd be double counting the overlap. Need to take into account that predictors may be correlated to each other.

  • If the predictors are correlated, they may ‘overlap’ in accounting for the variance in the Y 

  • You want to find out the unique (orthogonal) contributions of each predictor; you want one predictor to tell you stuff and then the other one to tell you unique non-overlapping stuff

What makes a good set of predictors?

• Low correlations with each other 

• High correlations with criterion

• When predictors (X1, X2) are highly correlated with each other, adding X2 to your regression model won’t substantially increase the total proportion of variability accounted for in Y

Difference between R^2 and r^2

R^2 distinguishes the use of multiple predictor variables from the use of just one predictor variable (r^2)

Beta vs b

• b= slope 

• Beta (B) → the regression coefficient (the ‘b’) when the data set has been standardized (made into z-scores) 

  • B allows for standard comparison between predictors/ also useful when researchers use different scales to measure the same variable 

  • For every 1 standard deviation increase in X, the predicted value for Y changes in B standard deviations