correlation ex post facto · pdf filerobert s michael correlation & ex post facto...

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Robert S Michael Correlation & Ex Post Facto designs-1

Overview:Correlational & Ex Post Facto

(aka “causal-comparative”) Designs

Y520Strategies for Educational Inquiry

Robert S Michael


What is Correlational Research?

■ Researchers want to know if there is arelationship between the number of sciencecourses students take and score on theNational Assessment of Educational Progressscience test.• What does “relationship” mean?• What data should they gather to decide

whether a relationship exists?

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Describe the Relationship (a)


Describe the Relationship (b)

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What is a Relationship?

■ We often have questions about relationships.■ Scores on the NAEP are related to:

• Geographic region• Amount of time watching TV• Type of computer use• Teacher’s area of undergraduate

preparation.• Type of science class taken.


Scattergrams Reveal Relationships

■ A Scatterplot is a graph that uses a coordinateplane to show the relationship betweentwo variables.

■ The convention is to place the exogenous(antecedent or predictor) variable on the X-axis (horizontal) and the endogenous(predicted or outcome) variable on the Y-axis.

■ Each data point shows the xy pair of values fora case.

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Displaying RelationshipsScattergram: X-Y pairs for each case

■ Strong PositiveRelationship:IQ and SpellingTest Scores



■ StrongPositiveRelationshipwith line ofbest fit.

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■ StrongNegativeRelationship


Another Example: Per Capita Income &Charitable Contributions

■ Is there a relationship between the amount ofmoney earned and the amount of charitablegiving?

■ We can list the per capita income for eachstate and the average itemized charitablecontribution (1998).

■ Each state has values for two variables.■ A good way to see whether a relationship

exists is to use a scatter plot.

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Scattergram: X-Y pairs for each case

State Per Capita Income vsPer Capita Charitable Contriubtions, 1998

$1,000

$2,000

$3,000

$4,000

$5,000

$6,000

$7,000

$8,000

$19,000 $24,000 $29,000 $34,000 $39,000

Per Capita Income by State

Avg

Item

ized

Con

trib

utio

nsby

Sta

te

Moderate NegativeRelationship(r = - 0.26)


Scattergram: X-Y pairs for each case

No Relationship

Weight and IQ

0

20

40

60

80

100

120

140

160

0 50 100 150 200 250 300 350

Weignt (lbs)

IQS

core

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Summary:

■ Relationships vary along two dimensions:■ Strength

• Strong• Weak• None

■ Direction• Positive• Negative


We can calculate the degree ofrelationship mathematically:

■ One formula is the Pearson ProductMoment Correlation Coefficient --(“Pearson r”)

■ Another is the Spearman Rho, used tocalculate correlation for rank ordered data.

■ Several other formulas exist for specialsituations.

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Example: Pearson r

■ The table showseach student’s testscore and numberof absences.

■ First, look at thescatterplot andguess thedirection andmagnitude.

Student Absences ScoreAlicia 1 92Bobby 4 73Carlos 5 86Donna 8 58Eddie 2 98Frea 0 97Gloria 4 70Helena 7 65Ingrid 0 88John 2 82

mean 3.3 80.9std dev 2.79 13.79


Example: Pearson r

■ The relationship isnegative andlikely strong.

■ The relationshipappears to belinear.

Relationship between Absences and Test Score

50

60

70

80

90

100

110

0 2 4 6 8 10

Absences

Tes

tSco

re

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Example: Pearson r

■ Second, calculate themean and standarddeviation for eachcolumn of values.

■ The standarddeviation is thesquare root of theaverage of thesquared deviations ofthe raw scores fromthe mean.

Student Absences ScoreAlicia 1 92Bobby 4 73Carlos 5 86Donna 8 58Eddie 2 98Frea 0 97Gloria 4 70Helena 7 65Ingrid 0 88John 2 82

mean 3.3 80.9std dev 2.79 13.79


Example: Pearson r

■ Third, convert each rawscore to its correspondingstandard (z) score.(Eliminates the problem ofdifferent units for differentvariables).

■ Standard scores have amean of zero andstandard deviation of 1.

■ Each person’s standard z-score shows theirdistance – above or below– the mean.

Student zabsences zscoreAlicia -0.8241 0.8051Bobby 0.2508 -0.5730Carlos 0.6091 0.3699Donna 1.6841 -1.6609Eddie -0.4658 1.2402Frea -1.1824 1.1677Gloria 0.2508 -0.7906Helena 1.3258 -1.1532Ingrid -1.1824 0.5150John -0.4658 0.0798

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Example: Pearson r

■ Fourth, for eachstudent, multiply thestandardizedabsences times thestandardized testscore.

■ The result shown inthe right hand columnis known as the“crossproduct”

■ Sum thecrossproducts anddivide by the numberof cases, minus one.

Student zabsences zscore zabsences * zscoreAlicia -0.8241 0.8051 -0.6635Bobby 0.2508 -0.5730 -0.1437Carlos 0.6091 0.3699 0.2253Donna 1.6841 -1.6609 -2.7971Eddie -0.4658 1.2402 -0.5777Frea -1.1824 1.1677 -1.3807Gloria 0.2508 -0.7906 -0.1983Helena 1.3258 -1.1532 -1.5289Ingrid -1.1824 0.5150 -0.6089John -0.4658 0.0798 -0.0372

-7.7106 < -- Sum

-0.8567 <-- Pearson


Example: Pearson r

■ Formula for converting araw score to a z-score:

■ Conceptual formulas forPearson r:

■ Spearman’s Rho :

■ For large data sets, usebuilt-in functions in Excelor statistical program.

zXraw Mraw–

SDraw--------------------------------=

rzxzy

n 1–------------------=

rspearman 1 6 D2

N N2

1– -------------------------–=

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Correlational Study: Data Gathering

■ Decide on the variables you suspect may berelated.

■ Decide how to measure those variables■ Select (preferably with a probability method) a

sample■ Measure the variables■ You have ONE GROUP of subjects but TWO

or more variables.


Correlational Study: Data Analysis

■ Use scatter plots■ Be sure the relationship is linear■ Look for trends and outliers■ Calculate the Pearson r or use other

appropriate correlation formula■ Note the direction and strength of relationship

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Factors that “distort” our interpretation ofCorrelation

■ Linearity• The line of best fit for a linear relationship is a straight line.• A curvilinear relationship yields an r value that underestimates the

true strength of the relationship

■ Restricted or truncate range of values• The sample exhibits limited values for one or both variables.• In either case, the value obtained for r is an underestimate.

■ Extreme Groups• r is overestimated. E.g., if the sample contains only poor and

excellent readers, the correlation is overestimated.

■ Extreme Scores• Inflates r


Factors that “distort”: Curvilinearity

■ Linear and curvilinear (or nonmonotonic) relationships

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Factors that “distort”: Restricted Range

■ Restricted Range: In thisexample, the correlationbetween foot size andage is much lower if werestrict age to one yearrather than considermultiple years.

■ Can range be artificiallyrestricted without theinvestigator’sawareness?


Factors that “distort”: Truncated Range

■ Truncated Range:The correlationbetween gradesand ACT scores islikelyunderestimatedbecause onlystudents at the highend of the gradingscale take the ACT.

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Factors that “distort”: Extreme Groups

■ Extreme Groups: In thisexample only poor andexcellent readers wereselected. Thecorrelation with IQ is“optimistic.” Inclusion ofmiddle range readerswould provide a morerepresentative sampleand lower thecorrelation.


Factors that “distort”: Extreme Score

■ Extreme Scores havea greater effect whenthe sample size issmall. In this example,a sample with noobvious relationshipappears to exhibit alinear relationship,due to a singleextreme score.

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Summary: Factors that Distort

■ Note that with the exception of linearity, all ofthe conditions that can distort the correlationare the result of a sample that is notrepresentative of the population:• Restricted range• Truncated range• Extreme groups• Extreme scores


Correlational Study: Results

■ Report obtained correlations■ Discuss strength and direction■ Determine statistical significance (Fisher’s r to

z transformation)■ Evaluate what it means in the context of your

study.■ Causality

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Correlational Study: Causality?

■ Correlation indicates little or nothing about causality.■ Inferring causality from correlation is a common error.

B ➠ C ➠ XB ➠ Y

d.

A causes both X and YX ➟ A ➠ Yc.

Y causes XX ➟ Yb.

X causes YX ➠ Ya.

ExplanationSymbolsPossibility


Interpreting Correlational

■ Correlation indicates little or nothing aboutcausality.

■ Inferring causality from correlation is a commonerror.

■ How much of the change in one variable isrelated to the change in the other variable?• This is known as “common variance” or the

“coefficient of determination” (r2 ).• Unexplained variance = 1 - r2

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Causal Comparative Research

■ We now shift from describing two or morevariables in one group, to

■ Comparison (two or more groups).


Causal Comparative Research

■ Involves comparison of two or more groups on a singleendogenous variables.

■ The characteristic that differentiates these groups is theexogenous variable.

■ Causal comparative studies are also called ex postfacto because the investigator has no control over theexogenous variable. Whatever happened occurredbefore the researcher arrived.

■ We can never know with certainty that the two groupswere exactly equal before the difference occurred.

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Causal Comparative: Data Collection

■ You select two groups that differ on the (exogenous)variable of interest.

■ Next, compare the two groups by looking at anendogenous variable that you think might be influencedby the exogenous variable.

■ Define clearly and operationally the exogenousvariable.

■ Be sure the groups are similar on all other importantvariables.


Causal Comparative: Equating groups

■ Use subject matching■ Use change scores; i.e., each subject as own control■ Compare homogeneous groups■ Use analysis of covariance

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Causal Comparative: Data Analysis

■ Because we usually are dealing with samples, we useinferential statistical testing techniques:• T-test (two groups)• Analysis of variance• Chi-square for frequency data


Causal Comparative: Conclusions

■ Researchers often infer cause and effect relationshipsbased on such studies.

■ Conditions necessary, but not necessarily sufficient, toinfer a causal relationship:• A statistical relationship exists that is unlikely

attributable to chance variation• You have reason to believe the supposed

exogenous variable preceded the endogenous.• You can, with some degree of certainty, rule out

other possible explanations.

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Discuss the relationship (correlation)


Types ofQuasi - Experimental Designs (a)

■ Time-series designs.■ Equivalent time-series samples■ Equivalent samples, materials design■ Non-equivalent control group■ Counterbalanced designs

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■ Begin here 00 – 10 - 14

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