c82mcp diploma statistics school of psychology university of nottingham 1 designing experiments in...

C82MCP Diploma Statistics

School of PsychologyUniversity of Nottingham

1

Designing Experiments

• In designing experiments we:• Manipulate the independent variable and hold all other

conditions constant.• Any observed changes in the dependent variable can be

attributed to the manipulation of the independent variable.



2

Correlational Studies• Sometimes the manipulation of independent variables is only

part of what we do• Sometimes we are more interested in finding out whether two

variables are related to each other.• For Example:

• Are IQ and reading ability associated with each other• In other words,

• Do high IQ scores and high reading ability scores occur together and low IQ scores and reading ability scores occur together



3

Correlational Studies

• When we ask about the relationship between two variables we have a correlational design• The null hypothesis associated with correlational designs

is that there is no relationship between variables• The alternative hypothesis associated with correlational

designs is that there is a relationship between variables



4

Covariance - A Measure of Association

• Covariance is defined as the sum of the product of two variables

• The value of the covariance is closely related to the relationship between X and Y• The larger the absolute value of the covariance the

stronger the relationship between X and Y.• The sign of the covariance tells us about the kind of

association

Covariance =(x - x )(y - y )

N



5

Covariance - A Measure of Association

• A Problem With Covariance• When there is an association between two variables the

value of the covariance depends on:• The degree or size of the association• The ranges of the two variables

X Ys1 1 4 2s2 2 6 0s3 3 8 2Mean 2 6 1.33

(X X )(Y Y ) X Ys1 10 40 200s2 20 60 0s3 30 80 200Mean 20 60 133.33

(X X )(Y Y )



6

Pearson's Product-Moment Coefficient of Correlation

• If we divide the covariance of two variables by the standard deviation of each variable we can account for the differences between the ranges• Pearson's Product-Moment Coefficient of Correlation, r, does

precisely this

• or

r Covariance(x, y)

sdx .sdy

r (x x )(y y )N.sdx .sdy



7

Pearson's Product-Moment Coefficient of Correlation

• The absolute value of this coefficient always lies between zero and one regardless of the different ranges of the variables

• The sign of the coefficient tells us what kind of association we have

• There are tables of values that estimate the probability of getting a particular value for Pearson's product-moment coefficient.

• Using these we can test the null hypothesis of no association between two variables.



8

An Example of a Positive Correlation

• Suppose a researcher measured the reading age and IQ of six 8 year old children

• The results of the measurement are placed on a scatter graph.

IQ vs Reading Age

80

85

90

95

100

105

110

115

120

125

6 6.5 7 7.5 8 8.5 9 9.5 10 10.5



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An Example of a Positive Correlation

• Given these data:

• The critical value of r for N=6 is 0.811• For a correlation to be significant

• robserved rcritical

• For these data there is no evidence of a significant association between IQ and Reading Age.

Reading Age IQs1 7.9 101 3.6049s2 8.2 113 -0.7361s3 6.3 94 30.3669s4 9.4 116 7.5499s5 8.4 107 -0.0501s6 10 121 20.0979mean 8.37 108.67 10.1389s.d. 1.28 10.01

(X X )(Y Y )

r Covariance(x, y)

sdx .sdy

r 10.1389(1.28)(10.01)

0.791



10

An Example of a Negative Correlation

• Suppose a researcher measured the reading age and the emotional and behavioural difficulties of six 8 year old children

• The results of the measurement are placed on a scatter graph.

EBD Scale vs Reading Age

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20

6 6.5 7 7.5 8 8.5 9 9.5 10 10.5



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• Given these data:

• The critical value of r for N=6 is 0.811• For a correlation to be significant

• robserved rcritical

• For these data there is no evidence of a significant association between EBD and Reading Age.

An Example of a Negative Correlation

Reading Age EBDs1 7.9 11 -1.0199s2 8.2 9 -0.0289s3 6.3 17 -16.9119s4 9.4 7 -1.8849s5 8.4 6 -0.0849s6 10 3 -9.5029mean 8.37 8.83 -4.90556667s.d. 1.28 4.83

(X X )(Y Y )

r Covariance(x, y)

sdx .sdy

r 4.9056(1.28)(4.83)

0.793



12

Spearman's Rank Correlation Coefficient

• Spearman Rank Coefficient of Correlation is an adaptation of Pearson's product-moment coefficient of correlation for ranks.• For each variable we rank the scores associated with that

variable. • We calculate the covariance and the standard deviations

for the ranks. • Dividing the covariance by the two standard deviations

we get Spearman's Rank Coefficient of Correlation.



13

An Example of Spearman's Rank Correlation (Rho)

• For a correlation to be significant

• rhoobserved rhocritical

• For these data there is no evidence of a significant association between IQ and Reading Age.

R.A. Rank IQ Ranks1 7.9 2 101 2 2.25s2 8.2 3 113 4 -0.25s3 6.3 1 94 1 6.25s4 9.4 5 116 5 2.25s5 8.4 4 107 3 -0.25s6 10 6 121 6 6.25mean 3.50 3.50 2.7500s.d. 1.87 1.87

(X X )(Y Y )

rho Covariance(x, y)

sdx .sdy

rho 2.75(1.87)(1.87)

0.775



14

Summary of Bivariate Correlations

• There are two regularly used correlation coefficient• Pearson's Product Moment Coefficient

• A parametric correlation coefficient• Spearman's Rho

• A non-parametric correlation coefficient• Correlation coefficient indicate

• The degree of a relationship between two variables• The direction of the relationship between two variables



15

Multiple Correlations

• We may be interested in looking at the pattern of correlations between several variables

• Such patterns correlations are usually summarised in a correlation table.

*p<0.05

Reading Ability Verbal Ability IQReading Ability 1Verbal Ability .621* 1IQ .736* .433* 1



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• There are many different models of the relationships between these variables.

• For example• Reading Ability and Verbal Ability may be caused by IQ

• Reading Ability and Verbal Ability may cause IQ

Correlations and Causal Models

ReadingAbility

VerbalAbility

IQ

ReadingAbility

VerbalAbility

IQ



17

Correlations and Causal Models

• It is difficult to know, on the basis of the correlations, which model is the most appropriate.

• We do not know which variables are directly or indirectly causing relationships between the variables

• Even worse some of the correlations may be spurious, i.e. not ‘true’ correlations at all.



18

Partial Correlations

• A technique known as partial correlations has been developed that allows us to look at the relationship between two variables in the presence of a third (or many other) variables.

• The partial correlation examines the degree of a relationship when other variables are controlled for.

• Given the zero order correlations, rxy, ryz and rxz the partial correlation is defined as:

rxy.z rxy (ryz)(rxz)

1 ryz2 1 rxz

2



19

Partial Correlations

• Looking at the partial correlation between reading ability and verbal ability controlling for IQ we find:

rRV . IQ rRV (rRIQ)(rVIQ)1 rRIQ

2 1 rVIQ2

rRV . IQ 0.621 (0.736)(0.433)1 0.7362 1 0.4332

rRV . IQ 0.495



20

Comparing Zero Order & Partial Correlations

• The zero order correlation between reading ability and verbal ability is 0.621 and the partial correlation when we have taken into account of IQ is equal to 0.495.

• Looking at the relationship between reading ability and IQ, the zero order correlation is 0.736 and the partial correlation when we account for verbal ability is 0.661.

• The zero order correlation for verbal ability and IQ is 0.433. The partial correlation is -0.045.



21

Causal Models and Correlations

• Given this pattern of results, one plausible explanation seems to be that IQ and verbal ability "cause" reading ability.

• However, correlation cannot itself inform us about causes. • We must appeal to theoretical considerations to try and

understand the causal relationships between different variables.

ReadingAbility

VerbalAbility

IQ



22

Causal Models and Correlations

• Given the pattern of zero order and partial correlations shown above, there are alternative accounts which are equally plausible.

• There are three other models as far as the statistical methods are concerned

ReadingAbility

VerbalAbility

IQ

ReadingAbility

VerbalAbility

IQ

ReadingAbility

VerbalAbility

IQ

c82mcp diploma statistics school of psychology university of nottingham 1 designing experiments in...

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kind of association

correlational studies

absolute value

correlational designs

high iq scores

high reading ability

low iq scores

particular value