Relationship between Variables: Correlation and Regression

Question 1

We are interested in finding a way to represent association between scores.

Answer 1

association

Question 2

The Regression Line

first and most obvious way to summarize data where we are examining the relationship between two variables.

Answer 2

Scatterplot

Question 3

The Regression Line

We put one variable on the x-axis and another on the y-axis, and we draw a point for each person showing their scores on the _____

Answer 3

two variables.

Question 4

The Regression Line

When we want to tell people about our results, we don’t have to draw a lot of _____

Answer 4

scatterplots

Question 5

Children were asked to listen to a word and repeat it. They were then asked which of these 3 words started with the same sound.

Answer 5

X

Initial phoneme detection

Question 6

reading score, a standard measure of reading ability.

Answer 6

Y

British Ability Scale (BAS)

Question 7

We usually summarize and represent the relationship between two variables with a number

Answer 7

correlation coefficient

Question 8

We also calculate the ______ for this number, and we want to be able to find out if the relationship is statistically significant.

Thus, we want to know what is the _______ of finding a relationship at least this strong if the null hypothesis that there is no relationship in the population is true.

Answer 8

Confidence Intervals

probability

Question 9

a best fitting line used for prediction.

Answer 9

Line of best fit or Regression Line

Question 10

Predicting the_____ in Y as a function of the ______ in X.

Answer 10

variation

Question 11

how steep the line

Answer 11

slope

Question 12

the position or height of the line.

Answer 12

intercept

Question 13

By ____ we give the height at the point where the line hits the y-axis.

Answer 13

convention

Question 14

The height is called the ____or often just the_____. (or sometimes the constant)

Answer 14

y-intercept or intercept

Question 15

The intercept represents the expected score of a person who scored zero on the ______

Answer 15

x-axis variable.

Question 16

It is often the case that the intercept doesn’t make any sense. After all, no one usually scores____

Answer 16

scores 0 or close to 0.

Question 17

We can use the two values of______ to calculate the expected value of any person’s score on Y, given their score on X

Answer 17

slope and intercept

Question 18

formula for Expected Y score

Answer 18

Expected Y score = intercept + slope x (score on X)

Question 19

Where x is the x-axis variable. This equation is called the ______

Answer 19

regression equation.

Question 20

Making Sense of Regression Lines

thinking about the relationship between______ can be very useful.

Answer 20

two variables

Question 21

Making Sense of Regression Lines

We can make a____ about one score from the another score.

Answer 21

prediction

Question 22

Problem: if we don’t understand the scale(s), regression lines and equations are _____

Answer 22

meaningless

Question 23

When there is a relationship between two variables, we can _____ one from the other.

We can not say that one _____the other,

Answer 23

predict

explains

Question 24

The correlation coefficient

We need some way of making the scales have some sort of meaning, and the way to do this is to convert the data into _____

Answer 24

standard deviation units.

Question 25

Thus we could ask: “If the score on ___ is one SD higher, how many SDs higher would we expect the ____score to be?”

Answer 25

x y

Question 26

Talking in terms of SDs means that we are talking about _____

Answer 26

standardized scores

Question 27

Because we are talking about standardized regression slopes, we call it______

Answer 27

standardized slope.

Question 28

Correlation coefficient – a more important name for the ______

Answer 28

standardized slope.

Question 29

Where σx is the SD of the variable of the variable on the x -axis (the horizontal one) of the scatterplot, and σy is the SD of the variable on the y-axis (the vertical one), and r is the correlation.

Question 30

The letter r actually stands for ______, but most people ignore that because it is confusing.

Answer 30

regression

Question 31

if we know the slope we can calculate the correlation using the formula:

Answer 31

r = β x σx / σy

Question 32

Residual In correlation, we want to know how well the ______line fits the data That is, how far away the points are from the line.

Answer 32

regression

Question 33

The closer the points are to the ____ the stronger the relationship between the two variables. (how do we measure this?)

Answer 33

line

Question 34

When we had one variable and we wanted to know the spread of the points around the mean, we calculated the____

Answer 34

SD (σ)

Question 35

The square of the SD is the ____

Answer 35

variance

Question 36

We can do the same thing with our regression data, but instead of making d the difference between the mean and the score, we can make it the difference between the value that we would expect the person to have, given their score on the x-variable, and the score they actually got. We can calculate their predicted scores, using:

Answer 36

y = b0 + b1x

Question 37

for each person, we can therefore calculate their predicted BAS reading score, and the difference between their predicted score and their actual score. The difference is called_____

Answer 37

Residual.

Question 38

the difference between the score they got and the score we thought they would get based on their initial score

Answer 38

residual score

Question 39

if we want to calculate the equivalent of the variance, we need to ____ each person’s score.

Answer 39

square

Question 40

The value of the standardized slope and the value of the square root of the proportion of variance explained will___ be the same value.

Answer 40

always

Question 41

We therefore have two equivalent ways of thinking about correlation. The first way is the _____. It is the expected increase in one variable, when the other variable increases by 1 SD.

Answer 41

standardized slope

Question 42

_We therefore have two equivalent ways of thinking about correlation. The second way is the ______. If you square a correlation, you get the ______ in one variable that is explained by the other variable

Answer 42

proportion of variance

Question 43

Interpreting Correlations A correlation is both ____ and ____

Answer 43

descriptive and inferential statistics

Question 44

We can find the probability estimate and we can also use it to describe the ____

Answer 44

strength of the relationship.

Question 45

strength of relationship

Answer 45

magnitude

Question 46

positive, negative, curvilinear etc.

Answer 46

direction

Question 47

r = 0.1 = small correlation * r = 0.3 = medium correlation * r = 0.5 = large correlation Note that these only really apply in what __, called Social and Behavioral sciences.

Answer 47

cohen's effect size

Question 48

Common mistake in interpreting correlations A correlation around 0.5 is a _____ * A correlation does not have to ____ 0.5 to be large. * If you have a correlation of r = 0.45, you have a correlation which is approximately ___ to a large correlation. * It’s not a ______ correlation just because it hasn’t quite reached 0.5

Answer 48

1. large correlation 2. exceed 3. equal 4. medium

Question 49

calculating the correlation coefficient Also known as Pearson Product moment correlation

Answer 49

Pearson Correlation Coefficient

Question 50

Pearson correlation coefficient developed by ____

Answer 50

karl pearson

Question 51

_____ correlation and makes the same assumptions made by other _____ tests.

Answer 51

Parametric

Question 52

pearson correlation coefficient is _______ data

Answer 52

Continuous and normally distributed data

Question 53

the moment is the length from the fulcrum multiplied by the weight on the lever.

Answer 53

physics

Question 54

the total moment is equal to the length from the center, multiplied by the weight.

Answer 54

seesaw analogy

Question 55

The same principle applies with _____

Answer 55

correlation

Question 56

We find the length from the center for each of the variables. In this case the center is the _____

Answer 56

mean

Question 57

So, we calculate the difference between ______ and _____ for each variable (these are the moments) and then we multiply them together (this is the product).

Answer 57

the score and the mean

Question 58

Because this value is____ on the number of people, we need to divide it by N.

Answer 58

dependent

Question 59

And because it is related to the _____, we actually divide by N-1 This is called _____, and if we call the two variables x and y

Answer 59

standard deviation covariance

Question 60

Finally, finding _____ is laborious, and we do not want to do it more than we have to.

Answer 60

square roots

Question 61

So instead of finding the square roots and then multiplying them together, it is easier to ______ together, and then find the square root.

Answer 61

multiply the two values

Question 62

importance scattergraph or plot: It will show us approximately what the correlation should be. So if it looks strong, ______, and our analysis shows it is -0.60. we have made a mistake.

Answer 62

positive correlation

Question 63

importance of scattergraph or plot It will help us detect any____ in our data, for example data entry errors.

Answer 63

errors

Question 64

importance of scattergraph or plot * It will help us get a feel of our ____

Answer 64

data

Question 65

The_____ for a statistic tell us the likely range of a value in the population

Answer 65

confidence intervals

Question 66

calculating confidence intervals Sampling distributions of correlation is _____

Answer 66

tricky.

Question 67

calculating confidence interval It is not symmetrical, which means we can’t _____ or _____ CIs in the usual way.

Answer 67

add and subtract

Question 67

calculating the pearson correlation transformation used which makes the distribution symmetrical.

Answer 67

* Fisher’s z transformation

Question 68

calculating the pearson correlation Used to calculate the CIs and then transform back to _____

Answer 68

correlations

Question 68

It is called a_____ because it makes the distribution of the correlation into a z distribution which is a normal distribution with a mean of 0 and SD of 1.

Answer 68

z transformation

Question 69

step _ Carry out Fisher’s transformation.

Answer 69

step 1

Question 70

step _ calculate the Standard Error

Answer 70

step 2

Question 71

step __ And now the CIs. We use the formula * CI = z’ + or – zα/2 x se

Answer 71

step 3

Question 72

Where zα/2 is the value for the ______ which includes the percentage of values that we want to cover.

Answer 72

normal distribution

Question 73

the value for the 95% confidence is (as always) ____

Answer 73

1.96

Question 74

Step _ Convert back to correlation.

Answer 74

step 4

Question 75

If we really want to know the_____ then we can convert the value for r into a value for t.

Answer 75

p-value,

Question 76

When we know the correlation we can also calculate the position of the _______

Answer 76

regression line.

Question 77

We can use the two values _______ to create a regression equation which will allow us to predict y _____ from x ______.

Answer 77

1. slope and intercept 2. (display behavior) 3. (desirability)

Question 78

If variables are both dichotomous (for example, yes/no, top, bottom) we can use the ____

Answer 78

Pearson correlation formula.

Question 78

f one of your variables is continuous and the other is dichotomous we can use the ___

Answer 78

Point Biserial Formula

Question 79

This is when one variable is categorical and has just two all-inclusive values. * Examples: Male/Female, Car owner/Non-Car owner, and so on.

Answer 79

Point Biserial Correlation

Question 80

Non-Parametric Correlations * Used when the data do not satisfy the assumptions of the Pearson Correlation because they are not normally distributed or are only ordinal in nature.

Answer 80

Spearman Correlation Kendall Correlation

Question 81

Three ways to deal with this problem:

Answer 81

Ignore it. It does not make a lot of difference. * Use the Pearson Formula on the ranks (although the calculation is harder than the Spearman formula). * Use a correction

Question 82

If we use a non-parametric test, such as a _____ we tend to lose power.

Answer 82

Spearman correlation

Question 83

Although we could be strict and say that rating data are strictly measured at an ordinal level, in reality when there isn’t a problem with the distributions, we would always prefer to use a ____

Answer 83

Pearson Correlation

Question 84

A_____ gives a better chance of a significant result.

Answer 84

Pearson correlation

Question 85

A curious thing about the _____ is how to interpret it.

Answer 85

Spearman

Question 86

We can’t say that it is the_____, that is the relative difference in the SDs, because the SDs don’t really exist as there is not necessarily any relationship between the score and the SD.

Answer 86

standardized slope

Question 87

We also can’t say that it is the ____ explained, because the variance is a parametric term, and we are using ranks.

Answer 87

proportion of variance

Question 88

All we can really say about the Spearman is that it is the Pearson correlation between the ___

Answer 88

ranks

Question 89

alternative nonparametric correlation, which does have a more sensible interpretation. (advantage: meaningful interpretation) Very rarely used however.

Answer 89

Kendall’s Tau-a (τ – Greek Letter)

Question 90

Kendall’s Tau-a is rarely used for two reasons:

Answer 90

Difficult to calculate if you do not have a computer. * It is always lower than a spearman correlation, for the same data (but the p-values are always exactly the same). * Because people like their correlations to be high, they tend to use it less.

Question 91

The fact that two variables correlate does not mean there is a causal relationship between them. * Though it is often very tempting to believe that there is.

Answer 91

Correlation and Causality

Question 92

* Correlation does not mean____, but ____ does mean correlation

Answer 92

causality

Question 93

In general, if one variable is a purely category-type measure, then correlation cannot be carried out, unless the variable is _____

Answer 93

DICHOTOMOUS

Question 94

Correlation is also a measure of association between_____

Answer 94

two variables

Question 95

What we can do with a nominal/categorical data is reduce the measured variable to nominal level and conduct a _____on the resulting frequency table.

Answer 95

chi-square test

Question 96

A lack of relationship is signified by a value close to ___

Answer 96

zero

Question 97

A value of zero however could occur for a ____

Answer 97

curvilinear relationship.

Question 98

Strength is a measure of the____

Answer 98

correlation.