Regression; Lec 8; Lab 4

Question 1

What is a coefficient?

Answer 1

A factor that makes up a particular property

Question 2

What does a significant r mean?

Answer 2

That the regression coefficient is also significant

Question 3

Which variable goes on X axis and which goes on Y axis?

Answer 3

Variable which ‘varies’ (IV) on the X-axis Variable measured (DV) on the y-axis

Question 4

If you want to predict university score from SAT scores - which is predictor variable and which is criterion variable?

Answer 4

SAT score is predictor variable

University score is criterion variable

Question 5

What is a negative correlation?

Answer 5

As one variable increases so the other decreases

Question 6

What does it mean when variables are said to covary?

Answer 6

That they have either a positive or negative correlation to one another

Question 7

What is the predictor variable?

Answer 7

Independent variable - used to predict an outcome (variable that varies)

Question 8

What are three things that are important to remember in regression?

Answer 8

1. Any set of data can have a regression line plotted
2. The significance of the correlation or regression tells us whether a real relationship exists
3. The correlation or the standard error of estimate tells us how accurate the regression equation is

Question 9

How should

rxy = cov(x,y)/SxSy

Where

Covxy = Σ(X - Xbar)(Y - Ybar)/N-1

be interpreted?

Answer 9

1. It is an indication of how closely the data points lie along the line of best fit (the regression line)
2. Like all stats requires a p value to determine whether the relationship is due to chance or is the product of a real relationship

Question 10

what is r^{2 =}

Answer 10

The proportion of the variance in the DV that is predictable from the IV.

Question 11

Regression produces eight different analyses:

1. Descriptive statistics,
2. correlations,
3. variables entered/removed,
4. Model summary,
5. ANOVA,
6. Coefficients,
7. Casewise diagnostics,
8. Residual Statistics

First we look at ‘casewise diagnostics’, which should we look at second? Why?

Answer 11

Model summary

This is related to the correlation

r is the Pearsonn correlation restated

r² is the coefficient of determination (a measure of relative variability) and indicates how much of the variation in the DV can be explained by variation in the IV

‘Std. Error of the Estimate’ is the std. error of prediction for all the values - this gives up a direct measure of the potential of our predictions using the regression equation. You compare this score to the SD of the criterion variable to get an indication of how useful the regression is (if the score in the regression is lower than that of the criterion variable score than it is useful)

Question 12

What is this the formula for?

Note: in this instance the denominator = N-2 because we are imagining 2 values involved in the prediction

Answer 12

Residual/error variance

Question 13

Regression produces eight different analyses:

1. Descriptive statistics,
2. correlations,
3. variables entered/removed,
4. Model summary,
5. ANOVA,
6. Coefficients,
7. Casewise diagnostics,
8. Residual Statistics

First we look at ‘casewise diagnostics’, then we look at ‘Model summary’, what should we look at third? Why?

Answer 13

Coefficients (factor that makes up a particular property)

1. Column ‘B’, the value idenitified as constant is the intercept (a in regression equation)
2. Slope/gradiant of the line = identified by name given to predictor variable (b in regression equation)

Question 14

H0 rho =

Answer 14

0

where rho is the population correlation coefficient

Question 15

r² can give us % predictable variance. Using the smoking and CHD example, where r² = .723² = .508, explain.

Answer 15

r = .713

r² = .723² = .508

Approximately 50% in variability of incidence of CHD mortality is associated with variability in smoking - NOTE: you cannot infer a cause and effect relationship

Question 16

This is the formula for the regression of the line.

How do you calculate b?

Answer 16

S_X² = variability of X

Question 17

What is the standard error of prediction?

Answer 17

Standard dev. of all predicted values minus the recorded value.

Question 18

How do we know if our prediction is better than just using the mean?

Answer 18

Total SS - residual SS

Question 19

What is the criterion variable?

Answer 19

Dependent variable (variable measured)

Question 20

What can badly affect Pearson’s r?

Answer 20

1. Data with outliers
2. Data that is not linear (e.g. curvilinear - a smooth curve of any shape)

Question 21

Cov_{xy = ?}

Answer 21

Cov_{xy =} Σ(X - Xbar)(Y - Ybar)/N-1

Question 22

There are two possible ways to predict - what are they?

Answer 22

1. Basic = difference from mean scores (average Y)
   
   1. Total sums of square
2. Regression = difference from the regression line (difference from Yhat)

Question 23

If you want to predict the incidence of CHD in population based on incidence of smoking - which is the predictor variable? Which is the criterion variable?

Answer 23

• Predictor variable X (IV) = average number of cigarettes smoked per head of population
• Criterion variable Y (DV) = incidence of CHD

Question 24

How do you remove a data point from analysis?

Answer 24

Data –> Select cases –> ‘Select Cases’ dialogue box –> ‘If condition is satisfied’ –> ‘If…’ –> move variable you want to exclude to box and then put ~ (tilda - should not equal) and then type value of outlier you want to remove (e.g. 12.45) –> continue –> OK

Then you must re-run the analysis you wanted to run.

Question 25

What are each of the values for the regression equation:

Predicted score = b x (predictor score) + a

Answer 25

Predicted score = (Slope/gradiant of the line from coefficients output) x (predictor score) + (Coefficients output column ‘B’, the value idenitified as constant is the intercept)

Question 26

What is the linear regression equation?

Answer 26

Yhat = predicted value of Y

X = smoking incidence in a country

b = slope of line - change in predicted Y due to one unit change in X

a = the intercept - the value of Yhat when X is at 0

Question 27

There are 4 columns in the casewise diagnostics output:

1. Std. Residual
2. Case number
3. Predicted value
4. Residual

What do each of them mean?

Answer 27

1. Std. residual = identifies how many stardard errors of prediction the selected data point is away from the regression line
2. Case number = Participant number
3. Predicted value = What SPSS (regression) predicted the value would be
4. Residual = The gap between the predicted score and the actual score

Question 28

How would you summarise what correlation summarises?

Answer 28

Correlation quanitifies the potential linear relationship between two variables; the supposition of linearity must be confirmed by inspection of the scatterplot

Question 29

What

Question 30

How would you run a regression analysis in SPSS?

Answer 30

Analyze –> Regression –> Linear –> Move predictor variable (IV) to ‘Independents’ box –> Move criterion variable (DV) to ‘Dependents’ box –> Statistics –> Descriptives –> Casewise diagnostics –> Continue –> OK

Question 31

Regression produces eight different analyses:

1. Descriptive statistics,
2. correlations,
3. variables entered/removed,
4. Model summary,
5. ANOVA,
6. Coefficients,
7. Casewise diagnostics,
8. Residual Statistics

Which should we look at first?

Answer 31

Casewise diagnostics.

• This will identify the data that can be considered outliers according to the ‘standard error of prediction’.
• If there are any outliers that are more than 3 std devs beyond the value predicted by the regression line, then they can be considered ‘extreme outliers’ and should be removed
• Recalculate the regression with the outlier removed

Question 32

How would you summarise what a scatterplot depicts?

Answer 32

A scatterplot depicts the nature of association between two variables in a graphical form

Question 33

What is the difference between correlation and regression?

Answer 33

Correlation allows you to establish whether two variables covary, but does not enable prediction (regression does)

Question 34

r = ?

Answer 34

degree to which X and Y vary together (covariability of X and Y) / Degree to which X and Y vary separately (Variability of X and Y separately)

Question 35

What is the intercept?

Answer 35

a = the value of Yhat when X is zero

Question 36

What correlation should you run if the data are non-parametric (e.g., curivlinear)?

Answer 36

Use Spearman’s correlation

Question 37

What should you do before you run Pearson’s r?

Answer 37

Produce a scatterplot to see if data is linear and check for outliers. If appropriate then remove outliers.

Question 38

What is this the formula for?

Answer 38

Standard error of estimate

It is the SD (sq root of variance - calculated by determing the variation between each data point relative to the mean) of predicted values and a common measure of accuracy of prediction

Question 39

What is the mathematical formula for Pearson’s R?

Answer 39

r_{xy =} cov(x,y)/S_xS_y

Question 40

Standard deviation

Answer 40

1. SD is a measure of spread
2. A low SD tells us that the data is clustered around the mean, while a high SD tells us that it is dispersed over a wider range of values
3. Used when the data is normally distributed
4. Tells us whether a data point is standard/expected, or unusual/unexpected
5. Represented by sigma

How to calculate:

1. calculate the mean
2. subtract the mean from each data point
3. Square each difference
4. Calculate the mean of the squared differences
5. Take the Square root

Question 41

How would you summarise what regression quantifies?

Answer 41

Regressions quantifies the degree of impact one variable has on another, thus enabling prediction

Question 42

What is a positive correlation?

Answer 42

As one variable increases so the other increases

Question 43

What is error variance in regression called?

Answer 43

Residual variance - it is the variability of predicted values

Question 44

b = slope of line - change in predicted Y due to one unit change in X

What is this known as?

Answer 44

Regression coefficient

Question 45

How do you run Pearson’s in SPSS?

Answer 45

Analyze –> Correlate –> Bivariate

Question 46

This is the formula for the regression of the line.

How do you calculate a?

Answer 46

Question 47

Describe Pearson’s Product-Moment Correlation Coefficient

Answer 47

The extent to which a criterion variable (Y) varies in conjunction with the predictor variable (X)