PSYC 1037: Research Skills II Dr Pam Blundell

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PSYC 1037 Research Skills IIDr Pam Blundell

• Lecture Two Relationships

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Outline

• Objectives
• Answers to last weeks worksheet
• Covariance and Correlation
• Theory
• Practice

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Objectives

• At the end of this lecture you should
• Know how to interpret scatterplots
• Understand the concepts of covariance and
  correlation
• Be able to calculate Covariance
• Be able to calculate various Correlation
  coefficients

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Worksheet

• A priori power

• Assuming a0.05, power of 0.9 gives us a d3.25
• So we need 59 people per group
• So 118 subjects!

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Worksheet

• Post hoc power

• We have a medium effect size
• d is less than 1, so power is less than 0.17
  17 chance of detecting a difference that is
  really there!

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Reading

• Howitt Cramer
• Stats book Chapters 6 7, 10
• SPSS book Chapter 7
• Field Chapter 4

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Correlational method

• Measures two or more characteristics of the same
  individual and computes the relationship between
  those characteristics
• Studies variance among organisms rather than
  variance among treatments

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Relationships

• Do variables co-vary?
• Change in one variable results in change in
  another variable
• We can look at scattergrams and make observations
• We need an objective way of calculating the
  correlation

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Research questions

• Is there a relationship between IQ and A Level
  points?
• Is there a relationship between IQ and height?
• Is there a relationship between IQ and the number
  of children you have?

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Plotting relationshipsThe Scattergram
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The Scattergram

• Each point on the scattergram represents a single
  individual
• Each axis is a different variable

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Scattergrams

• Enable us to look at a glance and see if there
  are any outliers
• Can show relationships

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Positive relationship
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Negative relationship
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No relationship
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Covariance

• Scattergram illustrates the relationship
  qualitatively
• Need to quantify that relationship how do the
  variable co-vary.

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Variance
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Covariance

• There is an error in this formula in Howitt
  Cramer (p62)

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Co-variance

• To calculate how two variables co-vary we
• Calculate the difference between each data point
  and the mean
• Calculate the cross-product value for each data
  point
• Add these up, and divide by N-1

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• If we plot the means on the graph, we can
  graphically see what the Covariance is!

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• and is given by
  the distance between each point and the mean
• You can see that with this positive correlation,
  the data points nearly all lie in two quadrants
  of the scattergram

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-

-

• If we multiply the X-value by the Y-value all
  the points on a positive correlation fall in the
  positive sections of the scattergram. If we
  added up these values, wed get a large positive
  value.

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Worked example
IQ A level points
101 20
113 25
108 24
119 30
99 21
96 18
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Scattergram
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IQ A level points
101 20 -5 -3 15
113 25 7 2 14
108 24 2 1 2
119 30 13 7 91
99 21 -7 -2 14
96 18 -10 -5 50
106 23 186
Means
Hence Co-variance 186/5 37.2
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Covariance

• Is not a standardised measure
• The value will depend upon the units used
• Doesnt allow us to compare the magnitude of the
  covariance between different things
• Therefore, we need to standardise the covariance

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Correlation coefficient

• The Pearson product-moment correlation
  coefficient
• -1 r 1

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IQ A level points
101 20 -5 -3 15
113 25 7 2 14
108 24 2 1 2
119 30 13 7 91
99 21 -7 -2 14
96 18 -10 -5 50
106 (8.9) 23 (4.3) 186
Hence Correlation r 37.2 / (8.9 x 4.3) 0.97
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Correlation

• r varies between -1 and 1
• Can work out probability values(or SPSS tells us)

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Warning

• Correlations tells us about LINEAR relationships

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Recap

• Scattergrams
• Covariance
• Correlation

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Do you want a break?

• Correlations in SPSS
• Biserial point correlations
• Non-parametric correlations

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Correlation in SPSS

• Enter the data into SPSS one variable in each
  column

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Correlation in SPSS

• Analyze ? Correlate? Bivariate

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SPSS Output
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Scattergrams in SPSS

• Graphs ? Scatter

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• Simple - when you have two variables
• Overlay - examine the effects of two or more
  variables on another variable
• Matrix - examine the relationship between all
  combinations of many different pairs of variables
  

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• Set the variables for the X-axis and the Y-axis

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Scattergrams in SPSS
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Exam performance and anxiety

• SPSS demonstration -- File ExamAnx.sav
• First, examine the data using Scattergram
• Y-Axis (the DV) performance ( of mark)
• X-Axis (the IV) exam anxiety (/100)
• Set Markers by relationship between the
  variables for different groups (e.g. gender)
• Overlay Scattergram, to compare effects of
  revision and anxiety on exam performance
• Now carry-out statistical test
• Analyse gt Correlation gt Bivariate

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Interpreting the output

• What does the diagonal represent?
• How are anxiety and revision time each related to
  exam performance?
• 1. Anxiety and performance, (r -.441, plt.001)
• 2. Revising and performance, (r .397, plt.001)
• 3. Anxiety and Revising, (r -.709, plt.001)
• What do 1, 2, and 3 mean?
• So can we say that anxiety causes poor
  performance?

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Causality

• Correlation does not infer causality
• Correlation can be caused by a third variable
  that we have not measured
• Correlation tells us nothing about the
  directionality of a relationship
• We have not manipulated any variables there is
  no independent variable

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R2

• R2 (Co-efficient of determination) tells you the
  amount of variability in one variable that is
  explained by the others
• How much variability in performance is explained
  by anxiety?
• (-.441)² 0.194 - multiply by 100 19.4
• 19.4 of the variance in performance is accounted
  by anxiety! 80.6 remains to be accounted by
  other variables!
• What about revision time?

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Correlation Summary

• Pearsons correlation coefficient (R)
• Requires parametric data
• Is extremely robust

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Non-parametric data

• There are other tests available to deal with
  non-parametric data
• Spearmans Rho
• Kendalls Tau
• Point-biserial correlations

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Spearmans Rho

• Use when there are violations of parametric
  assumptions/and or ordinal data (e.g. grades,
  first, upper second, second, third, and pass)

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Spearmans Rho

• First, rank the data!
• How do we rank a set of measurements/data

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Spearmans Rho

• Now, use the ranks instead of the raw numbers,
  in the same calculation as Pearsons.
• Example Is facial attractiveness associated
  with ratings of success?

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Rank the data
Face Attractive Attract_Rank Success Success_rank
1 3 3.5 2 2
2 2 1.5 2 2
3 5 6 4 7
4 2 1.5 3 4.5
5 6 7.5 4 7
6 4 5 4 7
7 3 3.5 2 2
8 6 7.5 3 4.5
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attract success
3.5 2.0 -1.0 -2.5 2.50
1.5 2.0 -3.0 -2.5 7.50
6.0 7.0 1.5 2.5 3.75
1.5 4.5 -3.0 0 0.00
7.5 7.0 3.0 2.5 7.50
5.0 7.0 .5 2.5 1.25
3.5 2.0 -1.0 -2.5 2.50
7.5 4.5 3.0 0 0
4.5 (2.2) 4.5 (2.5) 25.00
Hence Correlation r 25/ (8-1) x (2.2 x 2.5)
0.65
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Spearmans Rho in SPSS

• Similar to Pearsons
• CorrelateBivariate
• Make sure Spearmans Rho box is ticked

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Kendalls Tau

• Use when we have a smaller data set with a large
  number of tied ranks, i.e., if lots of scores
  have the same rank
• More sensitive than Spearmans Rho
• Used more often because in reality we are more
  likely to have a large number of tied ranks!
• See SPSS

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Point Biserial correlation

• When one of the variables is a dichotomous
  variable (categorical with only two categories)
• pregnant woman
• dead or alive
• male or female
• DONT use when there is really a continuum
  underlying the category (e.g. Pass/Fail).

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Point Biserial correlation

• Calculation simple same as Pearsons!
• Example The relationship between time spent away
  and gender of cats (file pbcorr.sav)
• code sexuality (0female, 1male)
• time (mins away)
• Carry out Pearsons correlation

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• What about the sign of the correlation?
• Ignore the sign - depends on your coding!
• How much of the variance in time spent chatting
  up is accounted by sexuality?
• R² (.378) ² .143, so 14.3 of the variance in
  time away from home is explained by gender.
• Do Descriptives - Why?

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Summary
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