Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3

Review: The Logic Underlying ANOVA There are k samples with which to estimate population variance X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 X 31 X 32. X 3n Sample 3

Review: The Logic Underlying ANOVA There are k samples with which to estimate population variance X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 X 31 X 32. X 3n Sample 3

Review: The Logic Underlying ANOVA There are k samples with which to estimate population variance X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 X 31 X 32. X 3n Sample 3

Review: The Logic Underlying ANOVA The average of these variance estimates is called the “Mean Square Error” or “Mean Square Within”

Review: The Logic Underlying ANOVA There are k means with which to estimate the population variance X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 X 31 X 32. X 3n Sample 3

Review: The Logic Underlying ANOVA This estimate of population variance based on sample means is called Mean Square Effect or Mean Square Between

The F Statistic MS error is based on deviation scores within each sample but… MS effect is based on deviations between samples MS effect would overestimate the population variance when there is some effect of the treatment pushing the means of the different samples apart

The F Statistic We compare MS effect against MS error by constructing a statistic called F

The F Statistic F is the ratio of MS effect to MS error

The F Statistic If the hull hypothesis: is true then we would expect: except for random sampling variation

The F Statistic F is the ratio of MS effect to MS error If the null hypothesis is true then F should equal 1.0

ANOVA is scalable You can create a single F for any number of samples

ANOVA is scalable You can create a single F for any number of samples It is also possible to examine more than one independent variable using a multi- way ANOVA –Factors are the categories of independent variables –Levels are the variables within each factor

ANOVA is scalable A two-way ANOVA: 4 levels of factor 1 X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn 3 levels of factor

Main Effects and Interactions There are two types of findings with multi-way ANOVA: Main Effects and Interactions –For example a main effect of Factor 1 indicates that the means under the various levels of Factor 1 were different (at least one was different)

Main Effects and Interactions 4 levels of factor 1 X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn 3 levels of factor

Main Effects and Interactions 4 levels of factor 1 X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn 3 levels of factor

Main Effects and Interactions 4 levels of factor 1 X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn 3 levels of factor

Main Effects and Interactions 4 levels of factor 1 X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn 3 levels of factor

Main Effects and Interactions A main effect of Factor 1 Factor Levels of Factor dependent variable means of each sample

Main Effects and Interactions There are two types of findings with multi-way ANOVA: Main Effects and Interactions –For example a main effect of Factor 1 indicates that the means under the various levels of Factor 1 were different (at least one was different) –A main effect of Factor 2 indicates that the means under the various levels of Factor 2 were different

Main Effects and Interactions 4 levels of factor 1 X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn 3 levels of factor

Main Effects and Interactions 4 levels of factor 1 X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn 3 levels of factor

Main Effects and Interactions 4 levels of factor 1 X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn 3 levels of factor

Main Effects and Interactions A main effect of Factor 2 Factor Levels of Factor dependent variable

Main Effects and Interactions There are two types of findings with multi-way ANOVA: Main Effects and Interactions –For example a main effect of Factor 1 means that the means under the various levels of Factor 1 were different (at least one was different) –A main effect of Factor 2 means that the means under the various levels of Factor 2 were different –An interaction means that there was an effect of one factor but the effect is different for different levels of the other factor

Main Effects and Interactions An Interaction Factor Levels of Factor dependent variable

Correlation We often measure two or more different parameters of a single object

Correlation This creates two or more sets of measurements

Correlation These sets of measurements can be related to each other –Large values in one set correspond to large values in the other set –Small values in one set correspond to small values in the other set

Correlation examples: –height and weight –smoking and lung cancer –SES and longevity

Correlation We call the relationship between two sets of numbers the correlation

Correlation Measure heights and weights of 6 people PersonHeightWeight a5’4120 b5’10140 c5’2100 d5’1110 e5’6140 f5’8150

Correlation Height vs. Weight 5’5’25’45’65’85’ Weight Height

Correlation Height vs. Weight 5’5’25’45’65’85’ a a Weight Height

Correlation Height vs. Weight 5’5’25’45’65’85’ a a b b Weight Height

Correlation Height vs. Weight 5’5’25’45’65’85’ a a b b, e c c d d ef f Weight Height

Correlation Notice that small values on one scale pair up with small values on the other 5’5’25’45’65’85’ a a b b, e c c d d ef f Weight Height

Correlation Scatter Plot shows the relationship on a single graph Like two number lines perpendicular to each other 5’5’25’45’65’85’ a a b b, e c c d d ef f Think of this as the y-axis Think of this as the x-axis

Correlation Scatter Plot shows the relationship on a single graph 5’5’25’45’65’85’10 a b cdef a b, e c d f Weight Height * * * * * *

Correlation The relationship here is like a straight line We call this linear correlation * * * * * *

Various Kinds of Linear Correlation Strong Positive

Various Kinds of Linear Correlation Weak Positive

Various Kinds of Linear Correlation Strong Negative

Various Kinds of Linear Correlation No (or very weak) Correlation y values are random with respect to x values

Various Kinds of Linear Correlation No Linear Correlation

Correlation Enables Prediction Strong correlations mean that we can predict a y value given an x value…this is called regression Accuracy of our prediction depends on strength of the correlation

Spurious Correlation Sometimes two measures (called variables) both correlate with some other unknown variable (sometimes called a lurking variable) and consequently correlate with each other This does not mean that they are causally related! e.g. use of cigarette lighters positively correlated with incidence of lung cancer

Next Time: measuring correlations