Matters arising

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Matters arising

• Summary of last weeks lecture
• The exercises
• Your queries

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The Pearson correlation (r)

• The PEARSON CORRELATION is a measure of a
  supposed linear association between two
  variables.

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Linear, but imperfect association

• If the scatterplot is elliptical in shape, a
  linear association is indicated.
• In psychology, all measurement is subject to
  random error.
• No association between measured variables is ever
  perfect.
• That is why the points do not all lie on a
  straight line.

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The Pearson correlation
Sum of products
Sums of squares
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Explanation

• The numerator of r is known as a SUM OF PRODUCTS
  (SP).
• It is the sum of products that captures the
  extent to which X and Y are associated, or
  CO-VARY.
• The sums of squares in the denominator merely
  constrain the range of variation of r.

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The sum of products captures covariation

• Points in the upper right quadrant have positive
  deviation products points in the lower left also
  have positive deviation products (a minus times a
  minus is a plus).
• Points in the other two quadrants have negative
  products.
• Since the positive products predominate, we can
  expect the covariance to be very large.
• The negative products are small the points are
  near the intersection of the mean lines.

Mean Actual Violence score
Mean Preference score
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An elliptical scatterplot

• This is fine.
• The elliptical scatterplot indicates that there
  is indeed a basically linear relationship between
  variable Y1 and variable X1.

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No association

• There is NO association between Z and Y.
• The high value of r is driven solely by the
  presence of a single OUTLIER.

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Anscombes rule

• When you examine a scatterplot (something you
  should ALWAYS do when interpreting a
  correlation), ask yourself the following
  question
• Would the removal of one or two points at random
  affect the basically ellipical shape of the
  scatterplot? If the shape would remain
  essentially the same, the value of r accurately
  reflects the association between the variables.

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Summary

• The Pearson correlation r is a measure of the
  strength of a SUPPOSED linear relationship
  between 2 variables.
• It is one of the most widely used of statistical
  measures but it is also one of the most misused.
  
• You should always try to see the scatterplot when
  interpreting a value of r.

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Exercise

• From the Violence data, obtain a scatterplot and
  calculate the Pearson correlation.

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Direction of causation

• When we measure and obtaining the correlation
  between two variables we nearly always do so
  because we believe that one variable X causes or
  influences the other Y.
• We have measured Exposure X and Violence Y
  because we have the hypothesis that X causes Y.

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The scatterplot of Y against X

• If we believe that X causes Y, we want to PLOT Y
  AGAINST X .
• We want a scatterplot with Y on the vertical axis
  and X on the horizontal axis.

Richard
John
Jim
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Ordering the plot
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The default graph
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The vertical scale

• Notice that the vertical axis begins at 3, rather
  than at zero.
• I like to see the whole scale on the vertical
  axis.
• Double-click on the graph to enter the Chart
  Editor.
• Double-click on the vertical axis to enter a
  dialog which will enable you to control the
  amount of the vertical scale that you can see.

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Ordering the full Y scale
Uncheck Auto and enter zero into the Custom slot.
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Final version
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Why do I like to see the entire scale on
the vertical axis?
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Beware!

• Modern computing packages such as SPSS afford a
  bewildering variety of attractive graphs and
  displays to help you bring out the most important
  features of your results. You should certainly
  use them.
• But there are pitfalls awaiting the unwary.

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Performance profiles

• We often want to see how mean performance varies
  (or not) over various treatment conditions.
• We might want to compare the performance of
  participants who have ingested different kinds
  (or dosages) of drugs with that of a comparison
  or control group.
• There is a set of methods known as Analysis of
  Variance (ANOVA) which enable us to do that.

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Ordering a means plot
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A picture of the results
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The picture is false!

• The table of means shows miniscule differences
  among the five group means!
• The graph suggested that there were vast
  differences among the means!

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A small scale view

• Only a microscopically small section of the scale
  is shown on the vertical axis.
• This greatly magnifies even small differences
  among the group means.

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Putting things right

• Double-click on the image to get into the Graph
  Editor.
• Double-click on the vertical axis to access the
  scale specifications.

Click here
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Putting things right

• Uncheck the minimum value box and enter zero as
  the desired minimum point.
• Click Apply.

Amend entry
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The true picture!
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The true picture

• The effect is dramatic.
• The profile now reflects the true situation.
• ALWAYS BE SUSPICIOUS OF GRAPHS THAT DO NOT SHOW
  THE COMPLETE VERTICAL SCALE!

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Your queries

• Several of you have e-mailed me asking how you
  fit a line graph to a scatterplot.
• Last week, I said that an elliptical scatterplot
  indicated that the relationship between the
  variables was basically LINEAR.
• So we want the best-fitting straight line through
  the points.
• This is known as the REGRESSION LINE.

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Drawing the regression line through the points
Choose Fit Line at Total.
To leave the Chart Editor, choose Close from the
Edit menu or double-click on the Viewer outside
the rectangle around the figure.
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Finding the value of r
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Hypothesis testing

• In HYPOTHESIS TESTING, a proposition known as the
  NULL HYPOTHESIS (H0) is set up.
• H0 is the NEGATION of your scientific hypothesis.
  
• So if our scientific hypothesis is that there is
  an association, H0 says theres NO association.

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The p-value

• To test H0, we gather our data and calculate the
  value of a TEST STATISTIC.
• If the null hypothesis is true, how probable
  would a value of our test statistic as extreme as
  ours have been?
• The answer is given by a probability known as the
  p-value.
• SPSS calls the p-value the Sig., i.e., the
  SIGNIFICANCE PROBABILITY.

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A significant result

• A SIGNIFICANCE LEVEL is a small probability
  accepted by convention as a criterion for a
  decision about a statistical test.
• Most commonly, the 0.05 significance level is
  accepted by psychologists.
• If the p-value of your test statistic is LESS
  than the 0.05 significance level, your result is
  said to be significant beyond the 0.05 level.

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The result
The p-value
Never report a p-value like this! Report the
p-value to 2 places of decimals if its less
than .01, use the inequality sign lt.

• Report this result as follows
• r(27) 0.89 p lt .01

Number of pairs value of r
p-value
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Lecture 9MORE ON ASSOCIATION
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We have shown that there is a strong
association between a childs violence and the
amount of violent screen material watched
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but have we really gathered evidence for
the hypothesis that exposure to screen violence
promotes actual violence?
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Remember

• CORRELATION
• does not necessarily mean
• CAUSATION

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One causal model

• The hypothesis implies this CAUSAL MODEL.
• The results are CONSISTENT with the hypothesis.
• The correlation may indeed arise because exposure
  to violence causes actual violence.

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Another causal model

• The childs violent tendencies towards and
  appetite for violence lead to his (or her)
  watching violent programmes as often as possible.
  
• This model is also consistent with the data.

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A third causal model

• NEITHER variable causes the other.
• Both are determined by the behaviour of the
  childs parents.

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The choice

• Does exposure cause violence (top model)?
• Does Violence lead to more exposure (middle
  model)?
• Are both exposure and violence caused by a third,
  background, variable (bottom model)?

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A background variable

• Perhaps neither Exposure nor Actual violence
  cause one another.
• Perhaps they are caused by a background parental
  behaviour variable.
• We have data on such a variable.
• The background variable correlates highly with
  both Exposure and Actual violence.

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Partial correlation

• A PARTIAL CORRELATION is what remains of a
  Pearson correlation between two variables when
  the influence of a third variable has been
  removed, or PARTIALLED OUT.

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Three variables

• Let X1, X2 and X3 be three variables.
• Let r12 be the Pearson correlation between X1 and
  X2.
• Let r(12.3) be the partial correlation between
  X1 and X2 when the covariation of each with X3
  has been removed.

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Partial correlation
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Explanation
Removes the influence of the third variable.
Rescales with new variances, so that the range is
as below.
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Obtaining a partial correlation
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The partial correlation

• The partial correlation fails to reach
  significance.
• Now that we have taken the background variable
  into consideration, we see that there is no
  significant correlation between Exposure and
  Actual violence.
• It appears that, of the three possible causal
  models, the third party model gives the most
  convincing account of the data.

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Levels of measurement

• There are three levels
• 1. The SCALE level. The data are measures on an
  independent scale with units. Heights, weights,
  performance scores and IQs are scale data. Each
  score has stand-alone meaning.
• 2. The ORDINAL level. Data in the form of RANKS
  (1st, 3rd, 53rd). A rank has meaning only in
  relation to the other individuals in the sample.
  A rank does not express, in units, the extent to
  which a property is possessed.
• 3. The NOMINAL level. Assignments to categories
  (so-many males, so-many females.)

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3. Nominal data

• NOMINAL data relate to qualitative variables or
  attributes, such as gender or blood group, and
  are merely records of CATEGORY MEMBERSHIP.
• Nominal data are merely LABELS they may take the
  form of numbers, but such numbers are arbitrary
  code numbers representing, say, the different
  blood groups or different nationalities. ANY
  numbers will do, as long as they are all
  different.

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A set of nominal data

• A medical researcher wishes to test the
  hypothesis that people with a certain type of
  body tissue (Critical) are more likely to show
  the presence of a potentially harmful antibody.
• Data are obtained on 79 people, who are
  classified with respect to 2 attributes
• 1. Tissue Type
• 2. Whether the antibody is present or absent.

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The research question

• Do more of the people in the critical group have
  the antibody?
• We are asking whether there is an ASSOCIATION
  between the variables of category membership
  (tissue type) and presence/absence of the
  antibody.
• This is the SCIENTIFIC hypothesis.

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The null hypothesis

• The NULL HYPOTHESIS is the negation of the
  scientific hypothesis.
• The null hypothesis states that there is NO
  association between tissue type and presence of
  the antibody.

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Contingency tables (cross-tabulations)

• When we wish to investigate whether an
  association exists between qualitative or
  categorical variables, the starting point is
  usually a display known as a CONTINGENCY TABLE,
  whose rows and columns represent the categories
  of the qualitative variables we are studying.
• Contingency tables are also known as
  CROSS-TABULATIONS, or CROSSTABS.

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The contingency table

• Is there an association between Tissue Type and
  Presence of the antibody?
• It looks as if the antibody is indeed more in
  evidence in the Critical tissue group.

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The null hypothesis

• The null hypothesis is the negation of our
  scientific hypothesis, namely, the statement that
  the two variables are INDEPENDENT.
• In other words, any differences in the relative
  incidence of the antibody in the different tissue
  groups have resulted from SAMPLING ERROR.

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Expected cell frequencies

• The pattern of the OBSERVED FREQUENCIES (O) would
  suggest that there is a greater incidence of the
  antibody in the Critical tissue group.
• But the marginal totals showing the frequencies
  of the various groups in the sample also vary.
• What cell frequencies would we expect under the
  independence hypothesis?

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Expected cell frequencies (E)

• According to the null hypothesis, the joint
  occurrence of the antibody and a particular
  tissue type are independent events.
• The probability of the joint occurrence of
  independent events is the product of their
  separate probabilities.
• We find the expected frequencies (E) by
  multiplying together the marginal totals that
  intersect at the cells concerned and dividing by
  the total number of observations.

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The expected frequencies

• To obtain, say, the value of E for the top left
  cell, multiply the intersecting marginal totals
  (36 and 22) and divide by 79 (the total
  frequency), obtaining
• (3622)/79 10.03 .
• In the Critical group, there seem to be large
  differences between O and E fewer Nos than
  expected and more Yess.

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The chi-square (?2) statistic

• We need a statistic which compares the
  differences between the O and E, so that a large
  value will cast doubt upon the null hypothesis of
  independence.
• Such a statistic is CHI-SQUARE (?2).

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Formula for chi-square

• The element of chi-square expresses the square of
  the difference between O and E as a proportion of
  E.
• Add up these squared differences for all the
  cells in the contingency table.

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The value of chi-square

• There are 8 terms in the summation, but only the
  first two and the last are shown in the
  calculation below.

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Degrees of freedom

• To decide whether a given value of chi-square is
  significant, we must specify the DEGREES OF
  FREEDOM df.
• If a contingency table has R rows and C columns,
  the degrees of freedom is given by
• df (R 1)(C 1)
• In our example, R 4, C 2 and so
• df (4 1)(2 1) 3.

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Significance

• SPSS will tell us that the p-value of a
  chi-square with a value of 10.655 in the
  chi-square distribution with three degrees of
  freedom is .014.
• We should write this result as
• ?2(3) 10.66 p .01 .
• Since the result is significant beyond the .05
  level, we have evidence against the null
  hypothesis of independence and evidence for the
  scientific hypothesis.

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Summary

• This week I extended my discussion of statistical
  association to the topic of partial correlation.
• A partial correlation can help the researcher to
  choose from different causal models.
• I also considered the analysis of nominal data in
  the form of contingency tables.
• The chi-square statistic can be used to test for
  the presence of an association between
  qualitative or categorical variables.

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Multiple-choice example
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Multiple-choice example
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Another example