Chapter 11 - Pearson\'s r test - worked example and logistics - parametic testing PDF

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Summary

Introduction to the Pearson's r test and worked example of the equation. Effect sizes explained and general correlational studies revised. Only parametric analyses for this test, so data must be interval or ratio data in order to get a reliable result....

Description

How To Tell If Two Variables Are Correlated 

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Looking for a relationship between two variables - how one variable influence another - positive correlation - as variable 1 increases, variable 2 increases - negative correlation - as variable 1 increases, variable 2 decreases Can calculate the Pearson’s correlation statistic to end up with an r value This value directly reflects the relationship of correlation - positive value r = positive correlation - negative value r = negative correlation r value ranges from -1 to 1 - value of 0 means there is no correlation - value of -1 means a perfect negative correlation - value of 1 means a perfect positive correlation Very rare to see a perfect positive or negative correlation - if one is achieved then the two variables may just be measuring the same thing The positive or negative value only says if the value is positive or negative, does not suggest strength in the direction or significance - the further away from 0, the stronger the significance r statistic will tell us if the correlation found is significant or not

Understanding The Role of Variability in A Pearson’s Correlation   

Does not compare experimental variance and random variance between or within groups Does work in a similar way though in terms of looking at different types of variability A line of best fit is often seen on the scattergrams of correlational studies - it is not just the slope of the line that is important, it is also the points scattered around the line

What Does the Pearson’s Correlation Do?    

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If all the raw data points are close to the line, the relationship is very consistence If the raw data points are all far from the line, the relationship has more variability The Pearson’s correlation looks at how much the data points change together - also known as covariance The more two variables covary and change in a similar way, the more covariance there will be - because in a positive correlation, both variables increase (change in same way) there is more covariance, a positive value of it - vice versa for negative correlation The covariance is the number calculated on the top part of the r equation

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The bottom part of the r equation looks at the variance within each variable that exists in isolation The end result of the equation is the covariance divided by the random variance (variance within variables) around the relationship - the more similar these numbers are, the stronger the calculated correlation will be (as a number divided by itself is 1)

Calculating And interpreting a Pearson’s Correlation Example     

Theories that the more intelligent people are, the more likely they are to be greater worriers 6 participants completed a task to measure their intelligence and a worrying questionnaire Intelligence tasks provides scores 5-12 - higher scores represent higher levels of intelligence Worrying questionnaires scores 1-10 - higher scores represent higher levels of worrying Hypothesis predicts there will be a positive relationship between intelligence and levels of worry

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Step 1: intelligence is labelled a, worry is labelled b Step 2: for each participant, multiply a scores by b scores Step 3: square each of the values and each of the b values Step 4: within each column find the sum of all scores Step 5: square the results from the sum of a and the sum of b - do not square results from sum of a squared or sum of b squared

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Step 6: all the above values can begin to be put into the equation

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Step 7: complete the multiplication sums within the brackets on the lower part of the equation

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Step 8: complete the remaining sums within the brackets, the multiplications on the top of the equation, and the subtractions on the lower part

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Step 9: calculate the subtraction on the top part of the equation and the multiplication on the lower part

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Step 10: calculate the square root sum

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Step 11: finally, calculate the division to calculate the Pearson’s r value r = .902

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The value is positive and therefore there is a positive correlation between the two variables

Is the Pearson’s Correlation Example Significant?  

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To find if the correlation is significant, the critical values table is needed from Appendix 3 of the Bourne book Four other pieces of information are needed: - Hypothesis one tailed or two? One tailed - What is the alpha level? .050 - What is the degrees of freedom? -N–2 -6–2 - df = 4 - what is calculated r value? +.902 Critical value can therefore be seen to be .7293 - this is the smallest r value still deemed to be significant - the calculated value must be equal to or larger than the critical value to be significant - .902 > .7293

What Should You Do with A Negative r Value?   

All the critical values are positive but negative correlations result in negative r values Just imagine in negative cases that all critical values have a negative sign in front of them Still want the negative r value to be more extreme than the critical value - therefore a negative r value needs to be a smaller number than a negative critical value

Interpreting and Writing Up a Pearson’s Correlation       

Need to include whether the correlation is significant and what direction the relationship is in If relationship is negative, don’t forget to put the (-) in front of the r value There is no need to include a (+) in front of a positive r value though If the result is not significant, then there is no need to interpret if the relationship was positive or negative APA standards are needed Descriptive statistics are not important Example results write up: - There is a significant positive correlation between the two variables (r (4) = .90, p < .050), this shows that individuals with higher levels of intelligence tend to report higher levels of worrying.

Effect Sizes for Pearson’s Correlation 

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Pearson’s r can be seen as an effect size without need for additional calculation - it exists on a continuous, standardised scale that runs from -1 (a perfect negative correlation) through to +1.0 (a perfect positive correlation) The r value can therefore be interpreted in the same way as an effect size (small, medium, or large) regardless of the sample size Small effect size would be greater than or equal .10 Medium effect size would be greater than or equal to .30 Large effect size would be greater than or equal to .50 As the r value for this study was .90, there was a large effect size

Which Variables Cannot Be Used for A Pearson’s Correlation?   

Correlations can only be used to analyse the relationship between two continuous variables of either interval or ratio data Interval and ratio data are parametric variables and the Pearson’s r is a parametric analysis Ordinal data is continuous, but it is non-parametric and therefore a Pearson’s r would not provide suitable analysis of this data...