Chapter_4_Field_2005 Correlation

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Chapter_4_Field_2005 Correlation

• If the cock crows on the dunghill, the weather
  will change or remain as it is (original German
  saying)?

Kikeriki
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What is a correlation?

• A correlation is a measure of a linear relation
  between variables (Clark, 2005,107)?
• Statistically, two measures express such a
  relation

Covariance Correlation coefficient(s)?
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Covariation

• Reminder Variation is the variability of a
  measure in a sample, e.g., height of students in
  this class
• Covariation is the relation between the variance
  of two variables, e.g., height and weight of
  students in this class.
• Q Are changes in one variable related to similar
  changes in the other variable?

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Measure of Covariance

• _ _
• Cov (x,y) ? (xi - x) (yi y)?
• N 1
• Compare measure of variance
• _
• Variance (s2) ? (xi - x)2
• N 1
• Instead of squaring the differences between x and
  
• _
• the mean x, as in the measure of variance, we
  multiply them with the differences between the
• _
• other variable y - y .

X weight Y height
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Problem with covariance

• The measure of covariance is not standardized,
  e.g., one cannot compare the covariance between
  two sets of data that are measured in different
  units.
• --gt convert covariance in a standard set of
  units.
• The standardized unit of measurement is the
  Standard Deviation SD of the mean.

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Pearson product-moment correlation coefficient r

• Correlation coefficient
• _ _
• r cov (x,y) ? (xi - x) (yi y)?
• sxsy (N 1)sxsy
• sxSD of the first variable
• sySD of the second variable
• --gt Due to standardization, r will always fall in
  between -1 and 1.
• Note that r is also used as a measure of the
  effect size of an experiment (see chapter 1)?

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Various kinds of correlation
r 1
rpos, weak
rpos, strong
r0
rneg, strong
r - 1
r nonlinear
r nonlinear
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Expl. Correlation analysis using SPSS (Field,
2005, pp 112, data set ExamAnxiety.sav)?

• 1. Visual inspection Scatterplot

Graphs--gt Interactive --gt scatterplot Exam
Performance y Exam Anxiety x Gender
Style (Fit None)?
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Output of scatterplotExam perform x exam anxiety

• Most students have high levels of anxiety
• No outliers
• Negative correlation
• No gender effect

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3D-scatterplots
Graphs --gt Interactive --gt Scatterplot
Use the following values

• 3D-scatterplot of exam performance plotted
  against exam anxiety and the amount of time spent
  revising

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Overlay scatterplots(not to be used with
'interactive graph'!)?

• In an Overlay scatterplot pairs of variables are
  plotted on the same axis
• Graphs --gt Scatter, choose 'Overlay'

Use swap pairs if necessary
Then 'define' the following pairs exam (Y)
anxiety (X) and overlay it with exam (Y)
revise (X) (use swap pairs if necessary
for bringing Y and X in the right order)?
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Overlay scatterplot

• Exam scores against both exam anciety and time
  spent revising

--gt pos rel between exam performance and exam
revision --gt neg rel between exam performance and
exam anxiety
Exam Per formance
Exam anxiety/Time spent revising
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Matrix scatterplotshows relation between all
combinations of different pairs of variables
perf (Y) anxiety (x)?
Perf (Y) rev time (X)?

• Graphs --gt scatter --gt Matrix --gt Define

Anxiety (Y) perf (X)?
Anxiety (Y) rev time (X)?
outlier
Rev time (Y) perf (X)?
Rev time (Y) anxiety (X)?
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Bivariate Correlationusing file Advert.sav
Q How are watching ads and buying the product
related? Task Bivariate correlation of both
variables
Analyze gt Correlate --gt Bivariate

• Bivariate Pearson's product-moment corr
• Corr coeff requires interval scale level
• Test for significance requires normal distribution

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Bivariate Correlationsusing ExamAnxiety.sav
Analyze --gt Correlate --gt Bivariate

• Neg corr between anxiety and perform, r -.441
• Pos corr between revision and perform, r .397
• Neg corr between anxiety and revision, r -.709

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The coefficient of determination R2

• When we want to know how much of the overall
  variability in the first variable can be
  determined by the second variable, we square
  Pearson's r.
• This coefficient of determination is written as
  R2
• Exp. r .871 (ad x packets bought) has a
• R2 .758
• 75 of the variance in the buying behavior can be
  accounted for by number of ads
• Caveat Still, R2 cannot be interpreted in a
  causal way.

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Spearman's correlation coefficient rsusing the
grades.sav data

• rs is used when the variables are not measured
  on an interval but on an ordinal scale. On the
  ordinal level, the assumption of a normal
  distribution does not have to be made.

• Expl Corr between Statistics Grades x Math grades

There is a pos correlation between Math and
statistics grades, rs.455
1-tailed test because of a directed hypothesis
(positive relation between math x stats)?
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Kendall's tau ??coefficientusing the grades.sav
data

• Kendall's tau ??is also a non-parametric
  correlation coefficient
• It is used for small data sets with large numbers
  of ranks.
• It is said to be a more accurate guess at the
  true correlation than Spearman's rs

Analyze --gt Corr --gt Bivariate--gt Kendall's tau
Same positive corr for Kendall's ? between
statistics and math grades as in Spearman's rs.
But ? is smaller.
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Biserial and point-biserial correlations

• Biserial and point-biserial correlations are used
  when one of the variables is only dichotomous

• The Point-Biserial corr coeff rpb is used when
  the underlying variable is truly dichotomous,
  e.g., male/female pregnant/not dead/alive, etc.
• The Point-Biserial corr coeff is Pearson's r

• The Biserial corr coeff rb is used when the
  underlying variable is dichotomous on the
  surface, e.g., having passed or failed in an
  exam, but underlyingly continuous, as expressed
  in the exact points earned.
• Rb cannot directly be calculated by SPSS

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Point-biserial Correlationusing the pbcorr.sav
data

• Q How is 'time roaming around' in a cat sample
  correlated with gender (male, female)?
• The Pearson coeff r .378
• R2 .143, i.e., gender accounts for 14.3 of the
  variance of cats' roaming around
• Whether the coeffient is pos or neg depends on
  which category is assigned which code. It
  reverses (from pos to neg) if instead of 'gender'
  the variable 'recode' is used (1female 0male)?

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Point-biserial Correlationusing the pbcorr.sav
data

• Analyze --gt Correlate --gt Bivariate --gt Pearson

Male 1 female 0
Whether the corr is pos or neg depends entirely
on the coding of the variables (malefemale)?
Male 0 female 1
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Computing rb (biserial r) from rpb
(point-biserial r)?

• If the underlying variable 'gender' is not truly
  dichotomous (because of some neutered male cats),
  then rb coefficient can be used, using the
  equation
• (E 4.4) Rb rpb ? (P1P2)?
• y
• where P1 is the proportion of cases that fell
  into category 1 and P2 of category 2 (male and
  female).

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Computing rb from rpb

• In the Menue 'Frequencies' we can obtain that
• P153.3 male
• P246.7 female
• Y the value of the normal distribution where P1
  stops and P2 begins
• In the Appendix A.1. , we find .3977 for .468 as
  the smaller portion and .532 as the bigger
  portion.
• Computing Equation 4.4. with y.3977, yields a rb
  of .475
• Rb is much higher than rpb!
• --gt It makes a difference, if the variable is
  truly dichotomous or continuous!

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Correlation and causalityThe standard disclaimer

• The correlation between two variables is an
  undirected relationship
• A causal interpretation is a directed
  relationship with the causing variable (causer)
  necessarily preceding and determining the caused
  variable (causee), in a meaningful way
• Problem 1 There could be a 3rd Variable
  mediating between the first two variables.
• Problem 2 There could be a complex interaction
  between the two variables, e.g., a positive
  feedback loop between exam anxiety and exam
  performance
• Therefore, never ever interpret a correlation as
  a causal story

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ExampleCorrelation ? Causation

• Correlation
• The number or storks and the number of human
  babies in a country are positively correlated.
• Causation?
• Does the birth rate go up because there are more
  storks? Do the storks bring the babies after all?

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Partial correlationusing the ExamAnxiety.sav data

• A partial correlation correlates two variables
  while keeping constant one or more additional
  variables
• In the examAnxiety data,
• Revision time is related to exam perform
• Revision time is related to anxiety
• In order to find out the true correlation between
  exam performance and anxiety, we have to partial
  out revision time

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Segmenting the variance

• Anxiety accounts for 19.4 of the variance of
  perfomance (r -.441)?
• Rev time accounts for 15.7 of the variance of
  performance (r .397)?
• Rev time accounts for 50.2 of the variance of
  anxiety (r -.709)?
• --gtparts of the 19.4 of which anxiety accounts
  for the performance variation may actually be
  accounted for by Rev time
• --gt Partial correlations may address the
  third-variable problem to some extent

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Diagram depicting partial correlations
Unique Variance explained by revision time
Exam performance
Revision time
Exam Anxiety
Unique variance explained by exam anxiety
Variance explained by both exam anxiety and
revision time
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Partial correlation between exam anxiety and exam
performance while controlling for revision time
Analyze --gt Correlate --gt Partial
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Partial Correlation

• The partial correlation between anxiety and
  performance is r -.2467 (R2 .06), having
  taken out revision time.
• (Originally it had been r -.441)?
• R2 has shrunk from 19.4 to 6

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Semi-partial (Part) correlations

• In a semi-partial (part) correlation, the effect
  of a third variable on one of the two variables
  is controlled, so that the unique corr between
  the two can be assessed.

• In a partial correlation, the effect of a third
  variable on both of the two variables is
  controlled

Revision
Revision
Exam
Anxiety
Anxiety
Exam
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Default Homework

• Answer the 'Smart Alex's tasks' in Chapter 4 (p.
  141)?

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Collective Homework

• Obtain a sample from this Statistics course with
  the following variables
• Weight, height, age, sex
• Explore the data with the commands for
  descriptive statistics you have learned so far
  Frequencies, mean, median, mode, SD, SE, range,
  variance, etc.
• Inspect the sample visually with histograms, box
  plot, scatter plots (simple, overlay, multiple)?
• Test for normal distribution (K-S-Test) and
  homogeneity of variances (Levene), split files if
  necessary
• Correlate all variable, see how high the corr
  are, if pos or neg
• If necessary, partial out variances