chapter 13 anova 1. relationship of statistical tests does this diagram make sense to you?

CHAPTER 13 CHAPTER 13 ANOVAANOVA

11

Relationship of Statistical Relationship of Statistical TestsTests

Does this Diagram Make Sense to You?Does this Diagram Make Sense to You?

Structural Equation Modeling

Structural Model

Multiple Regression

ANOVA

t ratio

Correlation

Measurement Model

Confirmatory Factor Analysis

Exploratory Factor Analysis

ANANALYSIS ALYSIS OOF F VAVARIANCERIANCE ANOVAANOVA

TESTS FOR DIFFERENCES AMONG TWO TESTS FOR DIFFERENCES AMONG TWO OR MORE POPULATION MEANSOR MORE POPULATION MEANS

σσ²²=S=S²²=MS=Variance=MS=Variance

MS=Mean MS=Mean ofof Squared Deviation Squared Deviation Ex of ANOVA Research: The effect of Ex of ANOVA Research: The effect of

temperaturetemperature on on recall. recall.

33

StatisticsStatisticsStandard Deviations and VariancesStandard Deviations and Variances

X X σ² = ss/N Popσ² = ss/N Pop 11 σ = √σ = √ss/Nss/N 22 44 s = √ss/df s = √ss/df SampleSample 5 5 s² = ss/n-1 or s² = ss/n-1 or ss/dfss/df

MSMS = = SS/dfSS/df

44

Effects of Temperature (IV) on Recall Effects of Temperature (IV) on Recall (DV)(DV)

One IV, 3 levels/Factors (50, 70, 90) and One IV, 3 levels/Factors (50, 70, 90) and one DVone DV

55

FFACTORIAL ANOVAACTORIAL ANOVA

66

MSMS betbet = SS = SS betbet / df / df betbet

MS MS withwith = SS = SS withwith /df /df withwith

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ANOVAANOVA

SS SS betbet = =ΣΣ(T²/n-G²/N)(T²/n-G²/N) SS SS withwith = =ΣΣssss SS SS totaltotal =SS =SS betbet + SS + SS withwith df df betbet = K-1 = K-1 df df withwith =N-K =N-K df df totaltotal= df = df betbet + df + df withwith

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Post Hoc Tests (Post Tests)Post Hoc Tests (Post Tests)

Post Hoc Tests are Post Hoc Tests are additional additional hypothesis hypothesis tests that are tests that are done done after after an ANOVA to an ANOVA to determine exactly determine exactly which which mean mean difference are difference are significant and which are not.significant and which are not.

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Post Hoc Tests (Post Tests)Post Hoc Tests (Post Tests) We use the PHT when we 1.reject the Null 2. We use the PHT when we 1.reject the Null 2.

there are 3 or more groups. 1.Tukeys, 2. there are 3 or more groups. 1.Tukeys, 2. Scheffe, Bonferroni, Duncan, LSD etc. Scheffe, Bonferroni, Duncan, LSD etc.

Tukey’s Honestly Significant Tukey’s Honestly Significant Difference(HSD) Test Difference(HSD) Test

q=Studentized Range Statistic

HSD= qHSD= q

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with / nMS

Coefficient of DeterminationCoefficient of Determination

If r = 0.80 then, r² = 0.64If r = 0.80 then, r² = 0.64This means 64% of the This means 64% of the

variability in the Y scores can variability in the Y scores can be predicted from the be predicted from the relationship with X.relationship with X. r=√r² r=√r² or or √r²=r

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Measuring Effect Size for Anova by Measuring Effect Size for Anova by Coefficient of DeterminationCoefficient of Determination

r²=r²=η²η² eta squared eta squared

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ProblemsProblems The data in next slide were obtained from an The data in next slide were obtained from an

independent-measures experiment designed to independent-measures experiment designed to examine people’s performances for viewing examine people’s performances for viewing distance distance of a 60-inch high definition television. of a 60-inch high definition television. Four viewing Four viewing distances were evaluated, distances were evaluated, 99 feet, feet, 1212 feet, feet, 15 15 feet, and feet, and 1818 feet, with a separate feet, with a separate group of participants tested at each distance. group of participants tested at each distance. Each individual watched a 30-minute television Each individual watched a 30-minute television program from a specific distance and then program from a specific distance and then completed a brief questionnaire measuring their completed a brief questionnaire measuring their satisfaction with the experience. satisfaction with the experience.

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ProblemsProblems One question asked them to One question asked them to rate the viewing rate the viewing

distance on a scale from distance on a scale from 1 (Very Bad 1 (Very Bad definitely definitely need to move closer or farther away) to need to move closer or farther away) to 7 7 (Excellent(Excellent-perfect-perfect viewing distance). The viewing distance). The purpose of the ANOVA purpose of the ANOVA is to determine is to determine whether there are any significant differences whether there are any significant differences among the four viewing distances among the four viewing distances that were that were tested. Before we begin the hypothesis test, tested. Before we begin the hypothesis test, note that we have already computed several note that we have already computed several summary statistics for the data in next slide. summary statistics for the data in next slide. Specifically, the tretment Specifically, the tretment totals (T) totals (T) and and SS SS values are shown for the entire set of data. values are shown for the entire set of data. 1616

ProblemsProblems9 feet 9 feet 12 feet 12 feet 15 fet 15 fet 18 feet18 feet

3 3 44 7 7 6 6 NN=20=20

0 0 33 6 6 33 GG=60=60

2 2 11 5 5 4 4 ΣΣX² X² =262=262

0 0 11 4 4 33 K=K=44

0 0 11 3 3 44

TT11=5 =5 TT22=10=10 T T33=25 =25 TT44=20=20

SSSS11=8 =8 SSSS22= 8 = 8 SSSS33=10 =10 SSSS44=6=6

MM11=1 =1 MM22=2=2 M M33=5 =5 MM44=4=4

nn11=5 =5 nn22=5=5 n n33=5 =5 nn44=5=5 1717

ProblemsProblems Having these summary values simplifies the Having these summary values simplifies the

computations in the hypothesis test, and we computations in the hypothesis test, and we suggest that you always compute these suggest that you always compute these summary statistics before you begin an summary statistics before you begin an ANOVA. ANOVA. We will set alpha at We will set alpha at α α =.05=.05

Step 1) Step 1)

HH00 : µ : µ11== µµ22=µ=µ33=µ=µ44 (There is no treatment effect.)(There is no treatment effect.)

HH11 : : (At least one of the treatment means is different.)(At least one of the treatment means is different.)

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Step 2Step 2

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ProblemsProblems A human factor psychologist studied A human factor psychologist studied

three computer keyboard designs. three computer keyboard designs. Three samples of individuals were Three samples of individuals were given material to type on a particular given material to type on a particular keyboard, and the number of errors keyboard, and the number of errors committed by each participant was committed by each participant was recorded. the data are on next slide.recorded. the data are on next slide. Set alpha at Set alpha at α α =.01=.01

2020

ProblemsProblemsKeyboard A Keyboard A Keyboard B Keyboard B Keyboard CKeyboard C

0 0 66 6 6 NN=15=15

4 4 8 8 5 5 GG=60=60

0 0 55 9 9 ΣΣX² X² =356=356

1 1 44 4 4

0 0 22 6 6

TT11=5 =5 TT22=25=25 T T33=30=30

SSSS11=12 =12 SSSS22=20 =20 SS SS33=14=14

MM11=1 =1 MM22=5=5 M M33=6=6 Is there a significant differences among the three computer keyboard designs ?

ProblemsProblems Step 1) Step 1)

HH00 : µ : µ11== µµ22=µ=µ33 (No differences between the (No differences between the

computer keyboard designs )computer keyboard designs )

HH11 : : (At least one of the computer keyboard designs is (At least one of the computer keyboard designs is

different.)different.)

2222

Step 2Step 2

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CHAPTER 14CHAPTER 14

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Chapter 15 Chapter 15

Correlation Correlation & &

RegressionRegression2525

X

Y

What is Correlation???What is Correlation??? Correlation measures the Correlation measures the strengthstrength and and

the the directiondirection of the relationship of the relationship between two or more variables.between two or more variables.

A correlation has three components:A correlation has three components:– 1.The 1.The strengthstrength of the coefficient of the coefficient– 2.The 2.The directiondirection of the relationship of the relationship– 3.The 3.The formform of the relationship of the relationship

1.The 1.The strengthstrength of the coefficient of the coefficient is is indicated by the absolute value of the indicated by the absolute value of the coefficient.coefficient.– The closer the value is to The closer the value is to 1.01.0, either , either

positive or negative, positive or negative, the stronger or the stronger or more linear the relationship.more linear the relationship.

– The closer the value is to The closer the value is to 00, , the the weaker or weaker or nonlinearnonlinear the relationshipthe relationship..

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X

Y

CorrelationCorrelation 2. The 2. The directiondirection of coefficient of coefficient is is

indicated by the sign of the correlation indicated by the sign of the correlation coefficient.coefficient.– A A positivepositive coefficient indicates that coefficient indicates that

as one variable (X) increases, so does as one variable (X) increases, so does the other (Y).the other (Y).

– AA negativenegative coefficient indicates that coefficient indicates that as one variable (X) increases, the as one variable (X) increases, the other variable (Y) decreases.other variable (Y) decreases.

– 3. The 3. The form form of the relationshipof the relationship The form of the relationship is The form of the relationship is linear.linear.

In correlation variables are In correlation variables are not identified not identified as independent or dependent as independent or dependent because because the researcher is measuring the one the researcher is measuring the one relationship that is mutually shared relationship that is mutually shared between the two variablesbetween the two variables– As a result, causality should not be As a result, causality should not be

implied with correlationimplied with correlation.. 2727

CorrelationCorrelation Remember, the correlation coefficient can only Remember, the correlation coefficient can only

measure a measure a linear relationshiplinear relationship.. A zero correlation indicates A zero correlation indicates no linear relationshipno linear relationship..

However, does not indicate no relationshipHowever, does not indicate no relationship..

a coefficient of zero rules out a coefficient of zero rules out linear linear relationship, relationship,

but a but a curvilinearcurvilinear could still exist.could still exist.– The scatterplots below illustrate this point:The scatterplots below illustrate this point:

05

1015

200

5

10

15

20

No Relationshipr = .0

05

1015

200

5

10

15

20

No Linear Relationshipr = 0

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The Correlation is based on a Statistic The Correlation is based on a Statistic Called Called CovarianceCovariance

Variance and Variance and CovarianceCovariance are used to are used to measure the quality of measure the quality of an item in a test.an item in a test.

Reliability and validity measure the quality of Reliability and validity measure the quality of the entire test.the entire test.

σ²=SS/Nσ²=SS/N used for one set of data used for one set of data

VarianceVariance is the degree of variability is the degree of variability

of scores from meanof scores from mean..

The Correlational MethodThe Correlational MethodSS,SS, Standard Deviations and VariancesStandard Deviations and Variances

X X σ² = ss/N Pop σ² = ss/N Pop 11 σ = √σ = √ss/Nss/N 22 44 s = √ss/df s = √ss/df 5 5 s² = ss/n-1 or ss/df Sample s² = ss/n-1 or ss/df Sample

SS=SS=ΣxΣx²-(Σx)²/N²-(Σx)²/N

SS=SS=ΣΣ(( x-x-μμ))²²

Sum Sum of of SquaredSquared DeviationDeviation from from MeanMean3030

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VarianceVariance

X X σ² = ss/N Popσ² = ss/N Pop 1 1 s² = ss/n-1 or ss/df Samples² = ss/n-1 or ss/df Sample 2 2 4 4 55

SS=SS=ΣxΣx²-(Σx)²/N²-(Σx)²/N

SS=SS=ΣΣ(( x-x-μμ))²²

Sum Sum of of SquaredSquared DeviationDeviation from from MeanMean

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CovarianceCovariance CorrelationCorrelation is based on a statistic is based on a statistic

called called CovarianceCovariance (Cov xy or S xy) = (Cov xy or S xy) = SP/N-1 SP/N-1 r=sp/√ssx.ssy -r=sp/√ssx.ssy - Correlation Correlation

CovarianceCovariance is a number that reflects is a number that reflects the degree to which 2 variables vary the degree to which 2 variables vary together. together.

Original DataOriginal Data X YX Y 1 31 3 2 62 6 4 44 4 5 75 7

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CovarianceCovariance

COVxy=SP/N-1COVxy=SP/N-12 ways to calculate the SP2 ways to calculate the SPSP= Σxy-(Σx.Σy)/N SP= Σxy-(Σx.Σy)/N ComputationComputation

SP= Σ(x-μx)(y-μy) SP= Σ(x-μx)(y-μy) DefinitionDefinition

SP requires 2 sets of dataSP requires 2 sets of dataSS requires only one set of SS requires only one set of datadata

The Correlational MethodThe Correlational Method CorrelationCorrelation is the degree to which is the degree to which events events

or characteristics vary from each otheror characteristics vary from each other

– Measures the Measures the strengthstrength of a relationshipof a relationship

–Does not Does not imply cause and effectimply cause and effect The people chosen for a study are The people chosen for a study are

its its subjects or participantssubjects or participants, , collectively called a collectively called a samplesample–The sample must The sample must be representativebe representative

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The Correlational MethodThe Correlational Method

Correlational data can be graphed and a Correlational data can be graphed and a “line of best fit” “line of best fit” can be drawn. can be drawn.

1- Pearson 1- Pearson CorrelationsCorrelations

2-Spearman2-Spearman

3-Point-Biserial Correlation3-Point-Biserial Correlation

4- Partial Correlation4- Partial Correlation

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Types of CorrelationTypes of Correlation In correlational research we use In correlational research we use

continues variables (interval or ratio continues variables (interval or ratio scale) forscale) for

1. Pearson Correlation (for linear 1. Pearson Correlation (for linear relationship).relationship).

If it is difficult to measure a variable If it is difficult to measure a variable on an interval or ratio scale then we on an interval or ratio scale then we use use

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Types of CorrelationTypes of Correlation 2.2.SSpearmanpearman Correlation Correlation SpearmanSpearman Correlation Correlation uses uses ordinalordinal or or

rank ordered datarank ordered data Spearman Correlation measures the Spearman Correlation measures the

consistency of a relationship consistency of a relationship (Monotonic Relationship)(Monotonic Relationship). . Ex. A teacher Ex. A teacher may feel confident about rank ordering may feel confident about rank ordering students’ leadership abilities but would students’ leadership abilities but would find it difficult to measure leadership on find it difficult to measure leadership on some other scale.some other scale.

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MonotonicMonotonic TransformationTransformation They areThey are rank ordered rank ordered numbers numbers (DATA), and use (DATA), and use ordinal ordinal scale(data) scale(data) examples; 1, 2, 3, 4, examples; 1, 2, 3, 4, or 2, 4, 6, 8.or 2, 4, 6, 8. Spearman Correlation Spearman Correlation can be can be used to measure the used to measure the degree of degree of Monotonic relationship Monotonic relationship between between two variablestwo variables..

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Ex. of Monotonic data Ex. of Monotonic data

X X YY

2222 8787

2525 102102

1919 1010

6 6 5 5

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Types of CorrelationTypes of Correlation 3.The Point-Biserial Correlation.3.The Point-Biserial Correlation. However, we can use both However, we can use both

continues and discrete continues and discrete variables(data) in variables(data) in The Point-The Point-Biserial Correlation. (can be a Biserial Correlation. (can be a substitute for two independent substitute for two independent t-test)t-test)

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3. The Point-Biserial Correlation3. The Point-Biserial Correlation The point-biserial correlation is used to The point-biserial correlation is used to

measure the relationship between two variables measure the relationship between two variables in situations in which one variable consist of in situations in which one variable consist of regular, regular, numerical scores numerical scores (non-dichotomies), (non-dichotomies), but the second variable has only two values but the second variable has only two values (dichotomies).(dichotomies).

We can also calculate this correlation from t-testWe can also calculate this correlation from t-test

r² = r² = Coefficient of Determination Coefficient of Determination which which measures the measures the effect size=deffect size=d

r² = t²/t²+dfr² = t²/t²+df r = √r²r = √r²

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4. A Partial Correlation4. A Partial Correlation

In special situations we can use In special situations we can use Partial Partial Correlations.Correlations.

Measures the relationship between two Measures the relationship between two variables while controlling the variables while controlling the influence of a third variable by holding influence of a third variable by holding it constant.it constant.

Ex. The correlation between Ex. The correlation between churcheschurches and and crime. crime. (third variable is population)(third variable is population)

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Correlational data Correlational data can be can be graphedgraphed and a and a “line of best “line of best fit” fit” can be drawncan be drawn

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Positive Correlation Positive Correlation

Positive correlation: Positive correlation: variables change in variables change in the same directionthe same direction

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Positive Correlation Positive Correlation

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Negative Correlation Negative Correlation

Negative correlation:Negative correlation: variables change in variables change in the opposite the opposite direction direction

4646

Negative CorrelationNegative Correlation

4747

No CorrelationNo Correlation

Unrelated:Unrelated: no no consistent consistent relationshiprelationship

4848

No CorrelationNo Correlation

4949


The magnitude (The magnitude (strengthstrength) of a ) of a correlation is also importantcorrelation is also important

–High magnitude =High magnitude = variables which variables which vary closely together; fall close to vary closely together; fall close to the line of best fitthe line of best fit

–Low magnitude =Low magnitude = variables which do variables which do not vary as closely together; loosely not vary as closely together; loosely scattered around the line of best fitscattered around the line of best fit

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The Correlational MethodThe Correlational Method Direction and magnitude of a correlation are Direction and magnitude of a correlation are

often calculated statisticallyoften calculated statistically– Called the Called the “correlation coefficient,” “correlation coefficient,”

symbolized by the lettersymbolized by the letter “r”“r”Sign (+ or -) indicates directionSign (+ or -) indicates directionNumber (from 0.00 to 1.00) indicates Number (from 0.00 to 1.00) indicates

magnitudemagnitude0.00 = no consistent relationship0.00 = no consistent relationship +1.00 +1.00 = = perfectperfect positive correlationpositive correlation -1.00 = -1.00 = perfectperfect negative correlationnegative correlation

Most correlations found in psychological Most correlations found in psychological research fall far short of research fall far short of “perfect”“perfect” 5151

The Correlational MethodThe Correlational Method Correlations can be trusted based on Correlations can be trusted based on

statistical probabilitystatistical probability– ““Statistical significance”Statistical significance” means that the means that the

finding is unlikely to have occurred by finding is unlikely to have occurred by chancechanceBy convention, if there is less than a 5% By convention, if there is less than a 5%

probability that findings are due to probability that findings are due to chance chance ((pp < 0.05), < 0.05), results are results are considered considered “significant” “significant” and thought to and thought to reflect the larger populationreflect the larger population

– Generally, Generally, confidenceconfidence increases with the increases with the size of the sample size of the sample and the and the magnitude of magnitude of the correlationthe correlation 5252


Advantages of correlational studies:Advantages of correlational studies:

– Have highHave high external validityexternal validityCan generalize findingsCan generalize findings

– Can repeat (replicate) studies on other Can repeat (replicate) studies on other samplessamples

Difficulties with correlational studies:Difficulties with correlational studies:

– LackLack internal validityinternal validityResults describe but Results describe but do not do not explainexplain a a

relationshiprelationship5353

External & Internal ValidityExternal & Internal Validity

External ValidityExternal Validity

External validity addresses the ability to generalize External validity addresses the ability to generalize your study to other people and other situations.your study to other people and other situations.

Internal ValidityInternal ValidityInternal validity addresses the "true" Internal validity addresses the "true" causescauses of the of the

outcomes that you observed in your study. Strong outcomes that you observed in your study. Strong internal validity means that you not only have internal validity means that you not only have reliable measures reliable measures of your of your independent (predictors)independent (predictors) andand dependent variables (criterions) dependent variables (criterions) BUT a strong BUT a strong justification that justification that causallycausally linkslinks your independent your independent variables variables (IV) (IV) to your dependent variables to your dependent variables (DV). (DV). 5454

The Correlational MethodThe Correlational MethodPearsonPearson

r=sp/√ssx.ssyr=sp/√ssx.ssy Original DataOriginal Data X YX Y 1 31 3 2 62 6 4 44 4 5 75 7

SPSP requires 2 sets of data requires 2 sets of dataSSSS requires only one set of data requires only one set of data

df=n-2df=n-25555

The Correlational MethodThe Correlational MethodSpearmanSpearman

r=sp/√ssx.ssyr=sp/√ssx.ssy Original Data Original Data Ranks Ranks X Y X YX Y X Y 1 3 1 11 3 1 1 2 6 2 3 2 6 2 3 4 4 3 24 4 3 2 5 7 4 45 7 4 4

SPSP requires 2 sets of data requires 2 sets of dataSSSS requires only one set of data requires only one set of data

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Percentage of Variance Accounted Percentage of Variance Accounted for by the Treatment (similar to for by the Treatment (similar to

Cohen’s d) is known as Cohen’s d) is known as ωω² Omega ² Omega Squared Squared also is called also is called Coefficient of Coefficient of

Determination Determination next page

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22

2

tr

t df

Coefficient of DeterminationCoefficient of Determination

If r = 0.80 then, r² = 0.64If r = 0.80 then, r² = 0.64This means 64% of the This means 64% of the

variability in the Y scores can variability in the Y scores can be predicted from the be predicted from the relationship with X.relationship with X. r=√r² r=√r² or or √r²=r

5858

ProblemsProblems Test the hypothesis for the following n=4 pairs of Test the hypothesis for the following n=4 pairs of

scores for a correlation, scores for a correlation, α α =.01=.01

r=sp/√ssx.ssyr=sp/√ssx.ssy Original DataOriginal Data X YX Y 1 31 3 2 62 6 4 44 4 5 75 7

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ProblemsProblems

Step 1) Step 1) HH00 : : ρρ=0 =0 (There is (There is nono population population correlationcorrelation.).)

HH11 : : ρ≠ρ≠0 0 (There is a real (There is a real correlationcorrelation.).)

ΡΡ: probability or chances …: probability or chances …

We will set alpha at We will set alpha at α α =.01=.01

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STEP 2STEP 2

6161

ProblemsProblems Test the hypothesis for the following set of Test the hypothesis for the following set of

n=5 pairs of scores for a n=5 pairs of scores for a positive positive correlation, correlation, α α =.05=.05

Original DataOriginal Data X YX Y 0 20 2 10 610 6 4 24 2 8 48 4 8 68 6

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ProblemsProblems

Step 1) Step 1) HH00 : : ρρ≤0 ≤0 ((The population correlation is ((The population correlation is not positive.)not positive.)

HH11 : : ρρ>0 >0 (The population correlation is (The population correlation is positivepositive.).)

ΡΡ: probability or chances are…: probability or chances are…

We will set alpha at We will set alpha at α α =.05=.05

6464

Three Levels of Analysis for Three Levels of Analysis for Prediction/validityPrediction/validity

INPUTSINPUTSPROCESSESPROCESSESOUTCOMESOUTCOMES Ex. Ex. Stress Stress (INPUT)(INPUT) is an unpleasant psychological is an unpleasant psychological

(PROCESS)(PROCESS) that occurs in response to that occurs in response to environmental pressures environmental pressures e.g., job demands) e.g., job demands) and can and can lead to lead to withdrawal (OUTCOME).withdrawal (OUTCOME). 66

66

Bi-Variate Regression AnalysisBi-Variate Regression Analysis Bi-variate regression Bi-variate regression analysis extends analysis extends

correlation and attempts to measure the extent correlation and attempts to measure the extent to which a to which a predictor variable (X) predictor variable (X) can be used can be used to make a to make a prediction about a criterion prediction about a criterion measure measure (Y). (Y).

X Y

Bi-variateBi-variate regression uses a regression uses a linear linear model model to to predictpredict the the criterioncriterion measure.measure.The formula for the predicted score is:The formula for the predicted score is:

Y' = a + bX Y' = a + bX 6868

Bivariate RegressionBivariate Regression The components of the line of The components of the line of

best fit (Y' = best fit (Y' = aa + + bbX) are:X) are:–thethe YY-intercept or (-intercept or (aa) is ) is

ConstantConstant–the slope the slope (b)(b)–Variable Variable (X)(X)

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Bivariate RegressionBivariate Regression TheThe Y-interceptY-intercept is the average value of Y is the average value of Y

whenwhen X is zero.X is zero.–TheThe Y-interceptY-intercept is also called is also called constant.constant.–Because, this is the amount of Y that is Because, this is the amount of Y that is

constant or present when the influence constant or present when the influence of X is null (0).of X is null (0).

The slope The slope is average value of a is average value of a one unit one unit change in Y change in Y for a corresponding for a corresponding one unit one unit change in X.change in X.–Thus, the Thus, the slope represents the direction slope represents the direction

and intensity of the line.and intensity of the line.7070

Regression and PredictionRegression and Prediction

Y=bX+aY=bX+aRegression LineRegression Line

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http://en.wikipedia.org/wiki/File:Linear_regression.png

Bivariate RegressionBivariate Regression Line of Best Fit: Line of Best Fit: Y' = 2.635 + .204X Y' = 2.635 + .204X

With this equation a predicted score may be made With this equation a predicted score may be made for any value of for any value of X X within the range of data.within the range of data.

a=2.635 a=2.635 andand b=.204b=.204

01

23

4

High School Grade Point Average

0

1

2

3

4

0

2

First Year Grade Point Average

Y-intercept

2.635

Slope = .204

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Multiple Regression AnalysisMultiple Regression Analysis Multiple regression analysis is an Multiple regression analysis is an

extension of extension of bi-variate regression, bi-variate regression, in in whichwhich severalseveral predictorpredictor variables are variables are used to predict used to predict one criterion one criterion measure measure (Y).(Y).– In general, this method is In general, this method is

considered to be advantageous; considered to be advantageous; since seldom can an outcome since seldom can an outcome measure be accurately explained measure be accurately explained by one predictor variableby one predictor variable. Ex. . Ex. 3 3 aspects of aspects of personality (OCPD, personality (OCPD, Narcissistic, Histrionic) Narcissistic, Histrionic) and and DepressionDepression

YX 2

X 1

X 3

Y' = a + bY' = a + b11XX11 +b +b22XX22 +b+b33XX33

E

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Path Analysis Path Analysis is an extension of regression.is an extension of regression. In Path analysis the researcher is examining the ability of In Path analysis the researcher is examining the ability of

more than onemore than one predictor predictor variable (X) to variable (X) to explain or predict explain or predict multiple multiple dependent variables dependent variables (Y)(Y). .

– 2 aspects of personality 2 aspects of personality (OCPD, Narcissistic) (OCPD, Narcissistic) and and

(Depression (Depression and Anxiety) and Anxiety)

Path AnalysisPath Analysis

E E

Y2

X1

X2

Y1

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Relationship of Statistical Relationship of Statistical TestsTests

Does this diagram make sense to you?Does this diagram make sense to you?

Structural Equation Modeling

Structural Model

Multiple Regression

ANOVA

t ratio

Correlation

Measurement Model

Confirmatory Factor Analysis

Exploratory Factor Analysis

chapter 13 anova 1. relationship of statistical tests does this diagram make sense to you?

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