Social Statistics: Correlation coefficient - PowerPoint Presentation

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Social Statistics: Correlation coefficientSlide2

Once you know the correlation coefficient for your sample, you might want to determine whether this correlation occurred by chance.

Or does the relationship you found in your sample really exist in the population or were your results a fluke?Or in the case of a t-test, did the difference between the two means in your sample occurred by chance and not really exist in your population.

Whether the correlation is significant

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If you set your confidence level at 0.05

Let’s assume that you collected your data with 100 different samples from the same population and calculate correlation each time. So, the maximum of 5 out of 100 samples might show a relationship when there really was no relationship (r=0)

Whether the correlation is significant

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Any relationship should be assessed for its significance as well as its strength

Pearson correlation measures the strength of a relationship between two continuous variablesCorrelation coefficient: rCoefficient of determination: r

2Significance is measured by t-test with p=0.05 (which tells how unlikely a given correlation coefficient, r, will occur given no relationship in the population)

The smaller the p-level, the more significant the relationship

The larger the correlation, the stronger the relationship

Correlation

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You have a sample from a population

Whether you observed statistic for the sample is likely to be observed given some assumption of the corresponding population parameter.Classical model for testing significance

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The classical model makes some assumptions about the population parameter:

Population parameters are expressed as Greek letters, while corresponding sample statistics are expressed in lower-case Roman letters:

r = correlation between two variables in the sample

(

rho) = correlation between the same two variables in the

population

A common assumption is that there is NO relationship between X and Y in the population:

= 0.0

Under this common

null

hypothesis in correlational analysis:

r

= 0.0 

Classical model for testing significance

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When the test is against the null hypothesis: r

xy = 0.0 What is the likelihood of drawing a sample with r xy

­ =0.0?The sampling distribution of r is

approximately

normal

(but bounded at -1.0 and +1.0) when N is

large

and distributes t when N is small

.

Classical model for testing significance

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The simplest formula for computing the appropriate

t value to test significance of a correlation coefficient employs the t distribution:  

The

degrees of freedom

for entering the t-distribution is

N - 2

T test for the significance of the correlation coefficient

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Example

Quality of Marriage

Quality of parent-child relationship

76

43

81

33

78

23

76

34

76

31

78

51

76

56

78

43

98

44

88

45

76

32

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Step1: a statement of the null and research hypotheses

Null hypothesis: there is no relationship between the quality of the marriage and the quality of the relationship between parents and childrenResearch hypothesis: (two-tailed,

nondirectional) there is a relationship between the two variables

T test for the significance of the correlation coefficient

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CORREL() and PEARSON()

Correlation coefficient

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r=0.393Slide12

Step2: setting the level of risk (or the level of significance or Type I error) associated with the null hypothesis

0.05 or 0.01

What does it mean?

on any test of the null hypothesis, there is a 5% (1%) chance you will reject it when the null is true when there is no group difference at all.

Why not 0.0001?

So rigorous in your rejection of false null hypothesis that you may miss a true one; such stringent Type I error rate allows for little leeway

T test for the significance of the correlation coefficient

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Step 3 and 4: select the appropriate test statistics

The relationship between variables, and not the difference between groups, is being examined.Only two variables are being used

The appropriate test statistic to use is the t test for the correlation coefficient

T test for the significance of the correlation coefficient

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=2.22

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Types of t test

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Step5: determination of the value needed for rejection of the null hypothesis using the appropriate table of critical values for the particular statistic

.From t table, the critical value=2.052 (two tailed,

0.05, df=27)T=2.22

If

obtained value>the critical value

reject null hypothesis

If obtained value<the critical value accept null hypothesis

T test for the significance of the correlation coefficient

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Step6: compare the obtained value with the critical

valueT Distribution Critical Values Table (Critical value r table)

compute the correlation coefficient (r=0.393)Compute df

=n-2 (

df

=27)

obtained

value: 0.393

critical value:

0.367

http

://www.gifted.uconn.edu/siegle/research/correlation/corrchrt.htm

T test for the significance of the correlation coefficient

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Step 7 and 8: make decisions

What could be your decision? And why, how to interpret?

obtained value: 0.393 > critical value: 0.349 (level of significance: 0.05)

Coefficient of determination is 0.154, indicating that 15.4% of the variance is accounted for and 84.6% of the variance is not.

There is a 5% chance that the two variables are not related at all

T test for the significance of the correlation coefficient

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Two variables are related to each other

One causes anotherhaving a great marriage cannot ensure that the parent-child relationship will be of a high quality as well;

The two variables maybe correlated because they share some traits that might make a person a good husband or wife and also a good parent;

It’s possible that someone can be a good husband or wife but have a terrible relationship with his/her children.

Causes and associations

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a correlation can be taken as evidence for a possible causal relationship, but cannot indicate what the causal relationship, if any, might be.

These examples indicate that the correlation coefficient, as a summary statistic, cannot replace the individual examination of the data.

A critique

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Exercise

To investigate the effect of a new hay fever drug on driving skills, a researcher studies 24 individuals with hay fever: 12 who have been taking the drug and 12 who have not. All participants then entered a simulator and were given a driving test which assigned a score to each driver as summarized in

the below figure.Explain whether this drug has an effect or not?

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