Correlation and Regression

Question 1

what is correlation a form of?

Answer 1

bivariate analysis
- relationship between 2 variables
focus on direction and degree

Question 2

what is a linear relationship?

Answer 2

for every increase in x, there is also an increase in why

Question 3

what are some examples of non linear relationship?

Answer 3

practice and performance… when learning a musical instrument, you are more likely to learn a lot more in the first year and your progress is likely to slow over time, eg

Question 4

T or F? Even when there is a non linear relationship, it makes sense to use correlation measures? Why

Answer 4

False

As you might get a correlation of value when in fact there is a U shape relationship between the data

Question 5

what are the rules of thumb on how big or small a correlation is?

Answer 5

small (.1 to .3)
medium (.3 to .5)
large (.5 to .7)

Question 6

what is r squared? what is it used for?

Answer 6

the correlation coefficient, squared
when you square the correlation coefficient, this gives you an estimate of the percentage of variance that is actually accounted for by your model - how much variance does your predictor account for?

Question 7

if your predictor accounts for 50% of the variance, what does this mean?

Answer 7

that 50% of the variation across subjects can be accounted for by the predictor you have

Question 8

what is variability?

Answer 8

how much a given variable varies from observation to observation - eg how much height in the class varies

Question 9

what is covariability?

Answer 9

how much two variables vary together
eg, if we take the class height and weight, as height increases (or decreases) how does that impact weight? positively, negatively or no relationship? do two variables vary together or independently of each other?

Question 10

what is the sum of squares used for? how is it calculated?

Answer 10

it calculates a rough estimate of variability…
SS = Σ(X - X(mean))^2
you take each individual’s height, and subtract the mean height from that and square it…. then sum it up for all observed numbers

Question 11

to measure the variability you use ____

to measure co variability you use ____

Answer 11

sum of squares; sum of products

Question 12

how is the sum of products calculated?

Answer 12

SP = Σ (X-X (mean)) x (Y - Y(mean))

Question 13

when will the sum of squares be identical to the sum of products?

Answer 13

when both variables are identical

Question 14

how do we calculate the pearson correlation coefficient?

Answer 14

SP
r = ———————
Square root of (SS of x by SS of y)

Question 15

what is the worded formula for calculating the pearson correlation coefficient?

Answer 15

r = covariability of X and Y/Variability of X and Y separately
calculating a ratio

Question 16

what happens if we have relatively low co-variability of X and Y compared to variability of X and Y separately?

Answer 16

we have a weak correlation

Question 17

what can drastically influence your correlation value?

Answer 17

extreme scores or outliers

Question 18

what is regression towards the mean?

Answer 18

where an extreme score on one measure tends to be followed by a less extreme score on the other measure… as extreme scores are often due to chance, it’s extremely unlikely that the other value will also be extreme, eg if there is a really really rainy day, it is likely that the following day will not be as rainy

Question 19

what is an example of the regression towards the mean?

Answer 19

1 or 2 people might guess 10 coin flips correctly, and 1 or 2 people might correctly guess the number between 1 and 50, but it is highly unlikely to be the same people as the extreme scores of the people who got the coin flip correct are more likely to be followed by getting a value closer to the mean on the next variable

Question 20

what is the null hypothesis for correlation?

Answer 20

that the correlation in the population is zero

Question 21

what is asked when determining if the null hypothesis is to be rejected or accepted?

Answer 21

once r value has been calculated, we ask what is the probability of finding an r value this big if the real association in the population is zero? If this probability is small, we reject the null hypothesis

Question 22

what is the degrees of freedom?

Answer 22

the amount of participants (N) minus 2

Question 23

how is spearman’s correlation used? when is it used?

Answer 23

convert the data to ranks before calculating correlations…

used when asking the question are values that are high on one variable also high on the other variable?

Question 24

why would you use spearman’s correlation?

Answer 24

when you have non linear data… and you control for or eliminate outliers as values such as 15, 20, 23232 becomes values of 1, 2, 3.

Question 25

what is reliability?

Answer 25

the consistency of a measure…. does a measure or test return the same results each time?

Question 26

what does cronbach’s alpha measure?

Answer 26

reliability

Question 27

what does cronbach’s alpha require?

Answer 27

at least 3 items or scales?

Question 28

how is cronbach’s alpha calculated?

Answer 28

by averaging covariance of item pairs divided by the total variance

Question 29

T or F? the value of cronbach’s alpha directly represents the proportion of reliable variance? Eg, value of .7 means 70% reliable variance?

Answer 29

True

Question 30

what is the rough rule of thumb for cronbach’s alpha?

Answer 30

excellent: equal to or greater than 0.9 
Good: 0.8 to 0.9
acceptable: 0.7 to 0.8 
questionable: 0.6 to 0.7 
poor: 0.5 to 0.6 
unacceptable: less than 0.5

Question 31

what is regression about?

Answer 31

predicting one variable from another

Question 32

what is the general formula for a perfect linear relationship? give example

Answer 32

Y = a + b X + e
Y = a + b x (IQ)

Question 33

what do all components of the general formula for a perfect linear relationship represent?

Answer 33

Y = outcome variable (what is being predicted) 
a = y intercept (value of y when x = 0)
b = slope (how much y changes whenever x changes) 
X = predictor variable 
e = error or residual term

Question 34

what are some assumptions about errors?

Answer 34

• errors are independent of one another
• normally distributed (if we were to plot all of our errors, it would roughly follow a normal distribution)
• homoscedastic (equal error variance for levels of predicted Y)

Question 35

how can we estimate regression model

Answer 35

least squares parameters estimates

Question 36

how do you calculate error or residual?

Answer 36

minus the predicted value from the observed value

Y - Y(predicted)

Question 37

how do you get the total squared error?

what is the formula?

Answer 37

calculate the error for every value then square them and sum them up
Σ (Y-Y(predicted))^2

Question 38

what is the sign for slope? how do you calculate it?

Answer 38

b.
SP
b = —————-
SSx

Question 39

what is the sign for intercepts how do you calculate it?

Answer 39

a.

a = Y(mean) - b x X(mean)

Question 40

what are other names for the intercept?

which one is the one they use in SPSS?

Answer 40

a
constant (this in spss)
y-intercept 
predicted valued of Y when x = 0

Question 41

what are other names for the slope?

which one is the one they use in SPSS?

Answer 41

b
rise over run
effect on Y for a unit increase (1) on predictor

Question 42

how do you measure the variance?

Answer 42

correlation score, squared

R^2

Question 43

what is r squared showing us?

Answer 43

the proportion of variance accounted for

Question 44

when do you use ANOVA test?

Answer 44

to determine whether the variance explained is significantly different from zero

Question 45

what does b (slope) indicate?

Answer 45

1. whether there is a relationship between X and y
2. whether that relationship is positive or negative
3. estimate of expected change in Y when x increases by 1
   DOES NOT indicate correlation (it is not the correlation)

Question 46

what do you have to do to the slope so that it matches the correlation? how do you do this?
what is this new coefficient called?

Answer 46

• translate it into a standardised form
• convert X and Y into Z scores and then do the regression
• standardised regression coefficient

Question 47

what regression diagnostics can be run?

Answer 47

histogram of residuals

residual plot - homoscedasticity

Question 48

what do we hope or expect to find when we run a histogram of residuals to be in the clear?

Answer 48

an approximately normal distribution

Question 49

what is a residual plot? what are we looking for?

Answer 49

• a scatterplot of residuals against predicted values to check for heteroscedasticity
• the absence of any systematic pattern supports the assumption of homoscedasticity

Question 50

T or F…. the standardised coefficient is not the same thing as the correlation (r)

Answer 50

False

Question 51

T or false
correlation tells us how tightly clustered the values are around the regression line but the line can have any sort of slope

Answer 51

True