Short description• Prereqs: I assume no prior knowledge of
stats
• This half day tutorial on statistical analysiswill introduce the most common statisticalapproaches used for analyzing behavioraldata, including t-tests, anovas, chi-squared, correlations, regressions and non-parametric tests.
• The focus will be on the appropriateapplication of these methods (when to usewhich test) rather than on their underlyingmathematical foundations. The introductionwill be presented in a very pragmaticproblem-oriented manner with examples ofhow to conduct the analyses using SPSS,interpret the program output and reportthe results. One goal of the tutorial will beto generalize the use of these tests …
• Won’t learn how to conduct the analyses, buthopefully enough background to help youbootstrap up (with help from r-tutorial or Dr.Google)
Structure of Tutorial
• Hour 1: Background and fundamentalunderpinnings to inferential statistics
• Hour 2: Tests for evaluating differences
• Hour 3: Tests for evaluating relationsbetween two (or more) variables
The Experiment
• An experiment should allow for asystematic observation of a particularbehavior under controlled circumstances.
• If we properly control the experiment,there should be minimal differencebetween the situations we create.
• Therefore any observed quantitative orqualitative difference must be due to ourmanipulation.
Two Types of Variables
• You manipulate the situations under whichthe behavior is observed and measured.The variables you manipulate are yourindependent variables.
• You observe and measure a particularbehavior. This measurement is yourdependent variable
Two Types of Variables
• The independent variable is the variablethat will be manipulated and compared orthat conditions or governs the behavior.– Word length, sentence type, frequency,
semantic relationship, instruction types, age,sex, task, etc.
• The dependent variable is the behaviorthat will be observed and measured.– Reaction times, accuracy, weight, ratings, etc.
IV Terminology
• Factor is another word for independentvariable, e.g., word frequency or length
• Level- value or state of the factor.• Condition (treatments)- all the different
situations you create by combining thelevels of your different factors.
IV Terminology: Example
• Question: Does the frequency of anantecedent influence the ease ofprocessing an anaphor?– IV (factor) 1 Antecedent frequency with two
levels (frequent vs. infrequent)• Alone, this would produce two conditions
– IV (factor) 2 Anaphor type with two levels(repeated NP vs. Pronoun)
Combining levels factorially
Condition 4Pronoun - frequent antecedent
Condition 3Pronoun - infrequent antecedent
Condition 2Repeated NP - frequent antecedent
Condition 1Repeated NP - infrequent antecedent
• Each level of each factor is combined with eachlevel of each other factor, we get 4 conditions
• 2 levels x 2 levels = 4 conditions• 2 levels x 3 levels = 6 conditions
How to choose a stat• 3 issues determine what statistical test is
appropriate for you and your data:
– What type of design do you have?
– What type of question do you want to ask?
– What type of data do you have?
Design Issues• How many independent variables do you
have?• How many levels of the IVs?• Is your comparison within
(related/dependent/repeated measures) orbetween (unrelated/independent)?– Within: subjects (items) are tested in all levels
of an IV.– Between: subjects (items) are tested only one
level of an IV.
Types of Questions• Different questions are addressed with
different tests.– Are my two conditions different from one
another?– Is there a relationship between my two
factors?– Which combination of factors best explains
the patterns in my data?
What type of data do you have?
• Parametric tests make assumptions aboutyour data.– Normally distributed
– Independent **
– Homogeneity of variance
– At least interval scale **
• If your data violate these assumptions,consider using a non-parametric test.
Normality• Your data should be from
a normally distributedpopulation.– Normal distributions are
symmetrical ‘bell-shaped’distributions with the majorityof scores around the center.
– If you collect enough data, itshould be normallydistributed (Central LimitsTheorem)
• Evaluate normality by:– histogram– Kolmogorov-Smirnov test– Shapiro-Wilks’ test.
Data Assumptions
Homogeneity of Variance• The variance should not change systematically
throughout the data, especially not betweengroups of subjects or items.
• When you test different groups of subjects(monolinguals vs. bilinguals; test vs. control;trained vs. untrained), their variances should notdiffer.– If you test two corpora, the variance should not differ.
• Evaluate with– Levene’s test for between tests– Mauchly’s test of Sphericity in Repeated Measures
Data Assumptions
Independence• Data from different subjects (items) are
independent.• The observations within each treatment condition
must be independent.– Subjects not randomly assigned to a group– If observations in one condition a subset of
observations in another condition– correlated samples, such as a set of pre- and post-test
observations on the same subjects, are not independent• Some tests are specifically designed to deal with dependent
(within) data.
Data Assumptions
Types of Data• Nominal scale:
– Numbers represent qualitative features, not quantitative.– 1 not bigger than 2, just different; 1=masculine, 2 = feminine
• Ordinal Scale:– Rankings, 1 < 2 < 3 < 4, but differences between values not
important or constant; Likert scale data.• Interval Scale: like ordinal, but distances are equal
– Differences make sense, but ratios don’t (30°-20°=20°-10°, but 20°/10° is not twice as hot)
– e.g., temperature, dates• Ratio Scale: interval, plus a meaningful 0 point.
– Weight, length, reaction times, age
Data Assumptions
What type of data do you have?
• Parametric tests make assumptions aboutyour data.– Normally distributed
– Independent **
– Homogeneity of variance
– At least interval scale **
• Parametric tests are not robust to violationsof the starred assumptions
Data Assumptions
Jonckheeretrend
VarianceRatio (F)
Independent
Mann-Whitney
χ2
Independent Z-Independent t-VarianceRatio (F)
Independent
Spearman’srank correlationcoefficient
Page’sL trend
Wilcoxon
Sign
One-sampleproportions
VarianceRatio (F)
Related tOne-sample ZOnesample t
Product-momentcorrelationcoefficient(Pearson’s r)Linearregression
RelatedRelated
CorrelationK sampleTwo-SampleOne-Sample
Type of Research DesignTy
pe o
f Dat
a
Para
met
ricN
on-p
aram
etric
Miller, 1984
Picking a test
Between-subjects analyses
χ2Kruskal-Wallis
Between-subjectsANOVA
Three ormore
χ2Mann-Whitney
Independentsamples t-test
two
Nonparametric -nominal
Nonparametric - ordinal
Parametricscores
# ofconditions
Picking a test
Within-subjects analyses
noneFriedmanRepeatedMeasuresANOVA
Three ormore
noneWilcoxonDependentsamples t
two
Nonparametric -nominal
Nonparametric - ordinal
Parametricscores
# ofconditions
Picking a test
Decision Tree (Howell, 2004)
Goodness-of-fitChi-squared
Type of CategorizationOne Categorical Variable
Contingency TableChi-squared
Type of CategorizationTwo Categorical Variables
Qualitative(Categorical)
PearsonCorrelation
Primary interestDegree of
Relationship
Regression
Primary interestForm of
Relationship
Continuousmeasurement
Spearman's r s
Ranks
One predictor
MultipleRegression
Multiplepredictors
Relationships
Two-sample t
Mann-Whitney
Independent
RelatedSample t
Wilcoxon
Dependent
Two
One-wayAnova
Kruskal-Wallis
One Independentvariable
FactorialANOVA
Multipleindependent
variables
Independent
RepeatedMeasures
Friedman
Dependent
Multiple
Numberof Groups
Differences
Type of Question
QuantitativeMeasurement
Type of data
Picking a test
How do stats work and whydo we use them?
• We observe some behavior, of individuals,the economy, our computer models, etc.
• We want to say something about thisbehavior. We’d like to say something thatextends beyond just the observations wemade to future behaviors, past behaviors,unobserved behaviors.
Types of Statistical Analyses
• Descriptive statistics: summarizing anddescribing the important characteristics ofthe data.
• Inferential statistics: decide if a pattern,difference or relations found with a sampleis representative and true of the population.
Why go beyond descriptives?
170190
Brides’ mean heightMonks’ mean height
• The mean might make it look like monks are taller, butmaybe that that doesn’t represent the truth.
Q: Are monks taller than brides?
Is this difference REAL?• Descriptive difference
– Milk costs .69 at Plus and .89 at Mini-mall• Is this a real difference?
• Subjective difference• Is the difference important enough to me? Is it worth my while
to travel farther to pay less?
• Statistical difference– Is Plus generally cheaper than Mini-mall?
• How representative of the prices is milk?– Are all Plus stores cheaper than all Mini-malls?
• How representative of all Plus stores is my Plus?
• Inferential statistics help us answer these questionswithout the need for an exhaustive survey.
How to inferential statisticswork?
• Different statistical methods attempt to builda model of the data using hypothesizedfactors to account for the characteristics ofthe observed pattern.
• One simple model of the data is the MEAN
ERROR
The mean
1F
4E
0D
2C
3B
1A
# siblingsSubjects
Mean # siblings = 1.83
How well does the meanmodel the data?
Variance• Variance is an index of error between the
mean and the individual observations.
!
xi" x( )#
!
xi" x( )#
2
• Sum of error is offset by positive and negative numbers– Take the square of each error value
• Sum of squared errors (SS) will increase the more data you collect.– Large number ≠ bad estimate of # of siblings
• Divide the sum of squared errors by N-1• Variance (s2) = SS/N-1
Standard Deviation
• Variance gives us measure in unitssquared, so not comparable to directly tothe units measured.– If your data has a range from 1-100, you could
easily have a variance of > 100, which is notthat informative an index of error.
• Standard Deviation is a measure of howwell the mean represents the data.
!
s = SSN "1
Sampling• Sampling is a random selection of representative
members of a population.• If you had access to all members of a population then
you would not need to conduct inferential statistics tosee whether some observation generalizes to thewhole population.
• Normally, we only have access to a (representative)subset.
• Most random samples tend to be fairly typical of thepopulation, but there is always variation.
Standard Error• Variance and Standard Deviation index the
relationship between the mean of observationsand individual observations.– Comment on the SAMPLE
• Standard error is similar to SD but it applies tothe relationship between a sample mean and thepopulation.
• Standard errors give you a measure of howrepresentative your sample is of the population.– A large standard error means your sample is not very
representative of the population. Small SE means it isrepresentative.
!
MSE = S / N
Normality and SD
• The combination of measures of varianceand assumptions of normality underliehypothesis testing and inferential statistics.
• Statistical tests output a probability that anobserved pattern (difference or relationship)are true of the population.
• How does this work?
Normal Curves• Normal curves can be defined by their
mean and variance.• Z-distribution has mean = 0 & SD = 1• x: N (0,1)
95% of data points are within about 2 SD of mean
Normal Curves
• Given these characteristics of normalcurves, you can calculate what percentageof the data points are above or below someany value.
• This is a basic principle behind most of thecommon inferential tests.
Example: IQ scores
• X:N(x,s2)• x = 100• s2 = 256; s=16
• What proportion of thepopulation has an IQof less than 84??
10084 11668 132
50%34%
100 - (50 + 34) = 16
Standard normal transformation• You can calculate the proportion above or below
or between any (2) point(2) for any normal curve.
• First, calculate how many SD a value is frommean. Then look up value in table.
10084 11668 132
!
z(x) =x " x
"
s
!
z(108) =108 "100
16= 0.5
http://davidmlane.com/hyperstat/z_table.htmlhttp://www.statsoft.com/textbook/sttable.html#z
Normal curves as gateway toinferential statistic
• Given the distribution of normal curves, you can now identify thelikelihood that two means come from the same population.
• Imagine you want to know if jetlag results in poor test taking ability.You test some kids who just flew from Edinburgh to Chicago.
• You know the population mean is 100 with SD=16. The sample youdraw has a mean test result of 84.– What can we conclude?
• We know that 16% of the general population has an IQ of 84 or below,so you want to know the chances of drawing, at random, a samplewith mean IQ of 84.
• If the chances are p > .05 (or whatever cut off you want to use) you’dconclude that jetlag doesn’t have a reliable effect on IQ. If p < .05 youwould conclude that it does.– That is statistical significance!!!
Sources of Variability
• There are two sources of variability in your data.– Variability caused by subjects and items (individual
differences)– Variability induced by your manipulation
• To find a statistically significant effect, you wantthe variation between conditions to be greaterthan within conditions.
• Note: if you don’t have variance in your data (e.g.,cuz the program is deterministic and alwaysbehaves the same way) inferential stats might notbe for you.
Variability of results
High variability(between 2 conditions)
median
Medium variability
Low variability (differencebetween conditions is reliable andcan be generalized)
Comparing two means
• If you are interested in knowing whetheryour two conditions differ from one anotherAND you have only 2 conditions.
• Evaluates the influence of yourIndependent Variable on your DependentVariable.
Test options: 2 conditions
• Parametric data: the popular t-test– 1 sample– Independent pairs– related pairs
• Non-parametric equivalents– Related pairs– Independent pairs
1-sample t-test• Compares mean of sample data to theoretical population
mean.• Standard error used to gauge the variability between
sample and population means.– The difference between the sample mean and the hypothesized
population mean must be greater than the normal variance foundwithin the population to conclude that you have a significant effectof the IV.
– Standard error small, samples should have similar means; SElarge, large diffs in means by chance are possible.
!
t =X "µ
standarderror ofmeanµ=estimated pop mean
Parametric tests of differences
How big is your t?
• How large should the t-statistic be?• The probability of a difference reflecting a REAL
effect is determined by the characteristics of theparticular curve.– Z and t curves are normal curves.– F and χ2 are positively skewed curves.– P-values are sensitive to the size of the sample, so a t
of 2.5 might be significant for a large N but not for asmall N.
– Tables or statistical programs relate t-values, degreesof Freedom and test statistics.
– http://faculty.vassar.edu/lowry/tabs.html#t
Independent samples t-test
• Compares means of two independentgroups (between-design).
• Same underlying principle (signal-to-noise)based on standard errors.
• Bigger t-value --> larger effects
!
t =(X "Y ) " (µ
1"µ
2)
Estimateof SE of difference
between twosamplemeans
Parametric tests of differences
Related t-test
• Compares the means of two relatedgroups; either matched-pairs within-subjects designs.– Test the same person in both conditions.– Reduce the amount of unsystematic variation
introduced into the experiment.
Parametric tests of differences
Non-parametric tests
• Most non-parametric tests work on theprinciple of ranking the data.
• The analysis is conducted on the ranks,rather than the data itself.– We lose information on effect magnitude, thus
non-parametric tests are less powerful thanparametric counterparts.
– Increased chance of type-II error (falsenegative)
Non-parametric tests of differences
814613Sub4a/b
715410Sub3a/b
51128Sub2a/b
3916Sub1a/b
RankCond 2RankCond 1
Ranking the Responses
Non-parametric tests of differences
Non-parametric tests
• Mann-Whitney: 2 independent pairs test.– Operates on ranking the responses,
independent of condition membership.– Equal sample size not required.
• Wilcoxon sign ranked: 2 related pairs– rank the absolute value of differences from a
pair– Equal sample size required
Non-parametric tests of differences
Sign Test• Alternative to either the 2 related or 2
independent (with random pairing) non-parametric tests.
• Requires equal sample size.• Simply counts number of positive vs.
negative differences betweenconditions.
• If no difference, 50/50 split expected.• Calculate the P (n +)
Non-parametric tests of differences
ANOVA
• Analysis of Variance– Similar to t-test in that it also calculates a
Signal-to-noise ratio, F.• Signal = variance between conditions• Noise = variance within conditions
– Can analyze more than 2 levels of a factor.• Not appropriate to simply conduct multiple t-tests
because of inflation of p-value.– Can analyze more than 1 factor.– Can reveal interactions between factors.
Parametric tests for more than 2 conditions
1-way ANOVA• Analogue to independent groups t-test for 3 or more levels
of one factor.– A 1-way anova with 2 levels is equivalent to a t-test. P-values the
same: F=t2
143966748,512
Total
6081,776141857530,417
WithinGroups
,0008,97954609,048
2109218,095
BetweenGroups
Sig.FMeanSquare
dfSum ofSquares
Parametric tests for more than 2 conditions
ANCOVA• Analysis of Covariance
– If you have continuous variable that was notmanipulated but that might add variance,• like word frequency, subject age, years of
programming experience, sentence length, etc…– you can factor out the variance attributed to this
covariate.– This removes the error variance and makes a
large ratio more likely.– Available for independent and repeated
measures variantsParametric tests for more than 2 conditions
Factorial univariate ANOVA
• When you have more than one IV but theanalysis remains between subjects.
• This analysis allows you to test the ‘maineffect’ of each independent variable andalso the interaction between the variables.
• If you have multiple dependent variables,you can use multivariate anova.
Parametric tests for more than 2 conditions
Repeated Measures Anova
• Within subjects
• This analysis is appropriate for data fromjust 1 IV, multiple IVs, for mixed designsand can factor out covariates.
Parametric tests for more than 2 conditions
Main effects and interactions• Let‘s assume we have 2 factors with 2 levels
each that we manipulated in an experiment:
– Factor one: lexical frequency of words (high frequencyand low frequency)
– Factor two: word length (long words and short words)
• We measured reaction times for a naming task.
• In a repeated measures ANOVA we canpotentially find 2 main effects and 1 interaction
Interactions
• Main effect of word length (long words 350 ms, short words 250 ms)• No main effect of frequency (high frequency 300 ms, low frequency 300 ms)• Interaction between word length and frequency (i.e., frequency has a different influence
on long words and short words)
Main effects and interactions
200
400
300 300
0
50
100
150
200
250
300
350
400
450
short words long words
high frequency
low frequency
Interactions
Interactions
• Interactions indicate that independentvariable X influences the dependentvariable differently depending on the levelof independent variable Y.
– Interpret your main effects with theconsideration of the interaction.
– You can have an interaction with no maineffects.
Interactions
Types of interactions
• Antagonistic interaction: the twoindependent variables reverse each other’seffects.
• Synergistic interaction: a higher level of Benhances the effect of A.
• Ceiling-effect interaction: the higher level ofB reduces the differential effect of A.
Interactions
Antagonistic Interaction
0
1
2
3
4
5
6
7
A1 A2
B1
B2
main
Effect
of A
Interactions
• the two independentvariables reverseeach other’s effects.
Interactions
Synergistic Interaction
0
2
4
6
8
10
12
14
A1 A2
B1
B2
main
Effect
of A
• a higher level of Benhances the effect ofA.
Ceiling-effect Interaction
0
2
4
6
8
10
12
14
A1 A2
B1
B2
main
Effect
of A
Interactions
• The higher level of Breduces thedifferential effect of A.
Interpreting Interactions
• If you want to say that two factorsinfluence each other, you need todemonstrate an interaction
• If you want to say that a factor affects twoDVs differently, you need to demonstratean interaction.
Interactions
Non-parametric alternatives
• If you have 1 independent variable withmore than 2 levels (K levels):
• Kruskal Wallis independent test forbetween designs
• Friedman related samples test for withindesigns.
Non-parametric tests of more than 2 conditions
Pearson’s Chi-squared• Appropriate if:
– 1 Categorical Dependent Variable (dichotomous data)– 2 Independent variables with at least 2 levels
• Detects whether there is a significant associationbetween two variables (no causality)
• Assumptions:– Each person or case contributes to only one cell of the
contingency table.• Not appropriate for repeated measures.
– takes raw data, not proportions.– Values in all cells should be > 5 (else do fisher’s exact
test)
Non-parametric tests of more than 2 conditions
Hierarchical Loglinear Analysis
• If you have more than 3 IndependentVariables and are interested in the HigherOrder Interactions.
• Designed to analyze multi-way contingencytables
• Table entries are frequencies– Functionally similar to χ2
Non-parametric tests of more than 2 conditions
Summary for anovas• If you have 1 independent factor with K
levels between subjects: 1-way ANOVA• If you have a covarying factor and one or
more between subjects independentfactors, use univariate ANOVA
• If you have repeated measures design,with 1 or more manipulated factors, with orwithout a covariate or an additionalbetween subjects factor, use RepeatedMeasures
Question
• Are two conditions different?
• What relationship exists between two ormore variables?– Positively related: as x goes, so goes y.– Negatively related: whatever x does, y does the
opposite.– No relationship.
Correlations
Example of linear correlation
Record Sales (thousands)
4003002001000
Advert
sin
g B
ud
get
(thousand
s o
f pounds)
3000
2000
1000
0
-1000
Correlations
Covariance• An association is indexed by covariance.
– Are changes in one variable met with a similar or oppositechange in another variable?
• Variance (s2) = SS/N-1• SS =
!
xi" x( )#
2
– We squared the error scores when lookingfor variance within one variable.
– If interested in the association between two variables, we multiply the error scorestogether.
Correlations
Calculating Covariance
• If deviations from the mean go in the samedirections for both variables, you’ll get apositive number.
• If deviations from the mean go in oppositedirections (one negative, one positive) you’llget a negative number.
!
cov(x,y) =xi " x( ) yi " y( )#N "1
Correlations
Interpreting linear relations
• Correlation coefficient [r] = linear relationship between twovariables. Measures the amount of spread around animaginary line through the center.
• r2 = proportion of common variation in the two variables(strength or magnitude of the relationship). Proportion ofthe variance in 1 set of score that can be accounted for byknowing X.
• Outliers? A single outlier can greatly influence the strengthof a correlation.
Correlations
Effect of outliers
One approach to dealing with outliers is to see if they arenon-representative (i.e., at the far end of the normal distribution).If so, they should be removed.
Correlations
Types of correlations
• Bivariate correlation: between two variable– Pearson’s correlation coefficient for parametric data
(interval or ratio data)
• Partial correlation: relationship between twovariables while ‘controlling’ the effect of one ormore additional variables.
• Biserial correlation: when one variable isdichotomous (e.g., alive vs. dead, male vs. female)
Correlations
Drawing conclusions• Correlations only inform us about a
relationship between two or morevariables.
• Not able to talk about directionality orcausality.– An increase in X does not CAUSE an
increase in Y or vise versa. Cause could befrom unmeasured third variable.
Correlations
R2
• By squaring our test statistic, we can tellhow much of total variance in the data forvariable x is in common with variable y.
• R2 = .2382 = .056 = 5.6% of variance. (94%of variability still unaccounted for!)
• .7632 = 0.582 = 58%
Correlations
Non-parametric correlations
• Spearman’s Rho– For non-interval data– Ranks the data and then applies Pearson’s
equation to ranks.• Kendall’s Tau
– Preferred for small data sets with many tiedrankings.
Correlations
Regressions• Correlations detect associations between
two variables.– Say nothing of causal relationships or
directionality– Can’t predict behavior on one variable given a
value behavior for another variable
• With Regression models we can predictvariable Y based on variable X.
Regressions
Simple Linear Regressions
• A line is fit to the data (similar to thecorrelations line).– Best line is one that produces the smallest sum
of squares from regression line to data points.
• Evaluation based on improvement ofprediction relative to using the mean.
Regressions
Hypothetical Data
0
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0 5 10 15 20 25
outcome variable
Pred
icto
r v
aria
ble
MeanRegressions
Error from Mean
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outcome variable
Pre
dic
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aria
ble
MeanRegressions
Error from regression line
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outcome variable
Pred
icto
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aria
ble
Mean
Regressionline
Regressions
Regression Results
• The best regression line has the lowestsum of squared errors
• Evaluation of the regression model isachieved via– R2 = tells you % of variance accounted for by
the regression line (as with correlations)– F = Evaluates improvement of regression line
compared to the mean as a model of the data.
Regressions
Predicting New Values
• Equation for line:– Y - output value– X = predictor value– β0 = intercept (constant. Value of Y without
predictors)– β1 = slope of line (value for predictor)– ε = residual (error)!
Y = "0
+ "1Xi+ #
i
Regressions
Multiple Regression
• Extends principles of simple linearregression to situation with multiplepredictor variables.
• We seek to find the linear combination ofpredictors that correlate maximally with theoutcome variable.
!
Y = "0
+ "1Xi+ "
nXn
+ #i
Predictor 1 Predictor 2 Regressions
Multiple Regression, con’t
• R2 gives the % variance accounted for bythe model consisting of the multiplepredictors.
• T-test tell you independent contribution ofeach predictor in capturing data.
• Logistic Regressions are appropriate fordichotomous data!
Regressions
R2 & Adjusted R2 should be similar
Adjusted R2 = variance in Pop
Is the model an improvementover the mean or over a priormodel?
β = Change inoutcome resulting inchange in predictor
Tests that lineisn’t horizontal
Regressions
Improvement over meanImprovement over 1stblock
Units expressed as standarddeviation for better comparison
T-test givesimpression of whethernew predictorsimprove model
Degree each predictoreffects outcome ifeffects of otherpredictors held constant
Summary• You should know which of the common tests are
applicable to what types of data, designs andquestions.
• You should also have enough backgroundknowledge to help you understand what you readon-line or in statistics books.
• You should have an idea which tests might beuseful for the type of data you have.
Informative Websites• http://faculty.vassar.edu/lowry/VassarStats.html
• http://www.uwsp.edu/psych/stat/
• http://www.richland.edu/james/lecture/m170/
• http://calcnet.mth.cmich.edu/org/spss/index.htm for spssvideos!
• http://espse.ed.psu.edu/edpsych/faculty/rhale/statistics/Investigating.htm
• Or, just google the test you are interested in.
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