3 VARIABLE HYPOTHESES 10 CONTENTS Basics 3 Variable

3 VARIABLE HYPOTHESES 10

CONTENTS Basics 3 Variable Tests

Contents Main Effects Interactions Covariation BASICS

3 VARIABLES • 3 -variable testing involves two independent variables (IVs) and one dependent variable (DV) • Sometimes you are interested in both independent variables • Sometimes the second IV is one which you don’t want, but may be having an influence ‣ but you need to take into account

A THIRD VARIABLE

A THIRD VARIABLE Main effect 1 Main effect 2

A THIRD VARIABLE Interaction Main effect 1 Main effect 2

A THIRD VARIABLE Covariation Interaction Main effect 1 Main effect 2

MAIN EFFECTS • Each IV may have an independent effect on the DV ‣ each IV causes a different pattern of variance in DV • In principle we can have many IVs ‣ each takes a share of the variance in DV

INTERACTION • The effect of IV 1 on the DV depends on the value of IV 2

INTERACTION • The interaction is how much the effect of IV on DV is changed by IV 2. ‣ In this case, the effect of IV on DV changes by 0. 2 between the two groups of IV 2

COVARIATION • The 2 IVs may be related • Look at IV & DV: ‣ there are 2 routes IV DV and IV 2 DV • Direct route ‣ zero effect size • Indirect route ‣ some effect

COVARIATION • Overall we will see an effect of IV on DV, but it isn’t a true effect: • It is confounded by IV 2

COVARIATION • These two situations show the same overall relationship between IV and DV • In the first case we think that IV affects DV, but the second case, we see that this is due to IV 2

THE EFFECTS • Main effects: ‣ usually interesting ‣ worth measuring • Interaction: ‣ sometimes interesting ‣ only worth measuring if you know what it means • Confounds: ‣ always important ‣ not worth measuring (we are focussed on the DV) ‣ but must always be considered

Contents General Linear Model Generalized Linear Model THE TESTS

DIFFERENT TYPES OF VARIABLE • When analysing 3 variables, we always treat Ordinal as if it was Interval

Contents 2 -way ANOVA ANCOVA Linear Regression THE TESTS (HISTORICAL)

VARIABLE TYPES • We will restrict ourselves to: ‣ DV is Interval • Each IV can be: ‣ Interval or Categorical IV 1 Type IV 2 Type Categorical Interval

3 -VARIABLE TESTS IV 1 Type IV 2 Type Categorical Interval

3 -VARIABLE TESTS IV 2 Type IV 1 Type Categorical Interval Categorical 2 -way ANOVA ANCOVA Interval ANCOVA multiple regression

Contents Switch 1: +ve or –ve Switch 2: on or off Switch 3: single group INTERACTIONS

A THIRD VARIABLE Interaction

INTERACTION • The effect of IV 1 on the DV depends on the value of IV 2 • In this example ‣ there is no main effect

INTERACTION • Overall ‣ IV has zero effect on DV • When IV 2=d 1, red dots, ‣ IV has +ve effect on DV • When IV 2=d 2, yellow dots, ‣ IV has –ve effect on DV

INTERACTION • The most visible sign of an interaction is that the lines have different slopes:

INTERACTIONS AS SWITCHES • Switch 1: plus or minus ‣ effect of IV on DV is either +ve or –ve • Switch 2: on or off ‣ effect of IV on DV is either on or off • Switch 3: on or off for 1 specific combination of values ‣ no effect of IV on DV except for female smokers

SWITCH 1: +VEOR –VE • IV 2=d 1(red) ‣ IV has –ve effect • IV 2=d 2 (yellow) ‣ IV has +ve effect

SWITCH 1: +VEOR –VE Note that the main effect sizes are zero

SWITCH 2: ON OR OFF • IV 2=d 1(red) ‣ IV has no effect • IV 2=d 2 (yellow) ‣ IV has effect

SWITCH 2: ON OR OFF • Note that the effect sizes are the same for: ‣ Main effect 1 ‣ Interaction ‣ But Main effect 2 is zero

SWITCH 3: ONE GROUP ONLY • IV 2=d 1 ‣ IV has no effect • IV 2=d 2 & IV=c 2 ‣ DV raised

SWITCH 3: 1 GROUP ONLY • Note that all the effect sizes are the same: ‣ Main effect 1 ‣ Main effect 2 ‣ Interaction

INTERACTIONS AS SWITCHES • Switch 1: plus or minus ‣ effect of IV on DV is either +ve or –ve • Switch 2: on or off ‣ effect of IV on DV is either on or off • Switch 3: on or off for 1 group (combination of categories) ‣ no effect of IV on DV except for female smokers

Contents COVARIATION

A THIRD VARIABLE Covariation

COVARIATION • The Covariation allows: ‣ IV 1 and IV 2 might be related • It is just another effect and has an effect size just the same ‣ -1 to 0 to +1

COVARIATION • We thought that there was an effect (0. 3) between IV 1 and DV)

COVARIATION • But actually there is a different effect: ‣ IV 1 IV 2 (0. 6) ‣ IV 2 DV (0. 5) • and no effect between IV 1 and DV • IV 2 is the confounding variable

TWO RULES • When we have a route that is a sequence of effects ‣ the effect size for the whole route is obtained by multiplying the individual effect sizes together • When we have a set of alternative routes: ‣ the effect size for the whole set of routes is obtained* by adding the individual route effect sizes together * this is an approximation

DIFFERENT EXAMPLE • Here we have two routes from IV 1 to DV: ‣ IV 1 IV 2 DV ‣ IV 1 DV • Their respective effect sizes are: ‣ 0. 6 x 0. 5 = 0. 3 ‣ -0. 3 = -0. 3 • The total effect size between IV 1 and DV is then: ‣ 0. 3 – 0. 3 = 0 This time, if we just look at IV 1 and DV, we see no effect, even though there is.

TOTAL &PARTIAL EFFECTS • The total effect size between IV 1 and DV is then: ‣ 0. 3 – 0. 3 = 0 ‣ this is called the total effect • The direct route on its own ‣ IV 1 DV (-0. 3) ‣ is called the partial effect

EFFECTS AS FLOW OF INFORMATION • effect size ‣ r 2 as proportion of variance explained • sequence of effects ‣ r all multiply together • parallel routes ‣ r all add together (approximately)

Contents Venn Diagrams 2 IVs Partial & semipartial EFFECT SIZES

EFFECTS &THE DV

EFFECTS &THE DV • There is a total variability in the DV • That is made up of: ‣ i. variability due to the IV ‣ ii. other variability

EFFECTS &THE DV • Variance ‣ uses deviations from the mean • Total variance ‣ deviations from the total sample mean • (a) deviation of any one point from total sample mean is made up of ‣ (1) deviation of its group mean from the total mean ‣ (2) plus deviation of it from its group mean overall mean group mean data point 1 2 a

EFFECT SIZE (BRIEFLY) • The deviation of any one point is made up of ‣ deviation of its group mean from the total mean ‣ plus deviation of it from its group mean • Deviation of all group means Effect size • Deviation all points from their group mean Residual

EFFECT SIZE (BRIEFLY) • Effect size: ‣ r and ranges from -1 to 0 to +1 • Here is a useful fact: ‣ r 2 (variance due to effect)/(total variance) ‣ proportion of variance due to effect ‣ “amount of variance explained” • And: ‣ amount of variance explained by different independent IVs just add together ‣ and can never exceed 100%

EFFECT SIZES • Here there are two IVs • IV 1 explains 0. 3 squared of the total variance ‣ 0. 3 x 0. 3 = 9% • IV 2 explains 0. 5 squared ‣ 0. 5 x 0. 5 = 25% • Together they explain 34% of total variance

EFFECT SIZE AS INFORMATION • We can think of the effect size as a measure of how much information flows from one variable to the next. • Then we can get much further

AN EASY WAY TO THINK OF EFFECTS • Venn diagrams

IV & DV • Each circle represents the variance of that variable • Where they overlap, there is an effect IV 1 DV IV 1 has strong relationship with DV

IV & DV The overlap illustrates the relationship between the variables. In the ANOVA terms: A = ssq(IV) B = ssq(Error) A+B = ssq(Total) B IV 1 DV Standardized effect = A/B Normalized effect = A/(A+B) A IV 1 has strong relationship with DV

2 IVS • Each circle represents the variance of that variable • Where they overlap, there is an effect IV 1 DV IV 1 has strong relationship with DV IV 2 has weak relationship with DV IV 1 has no relationship with IV 2

2 IVS A = ssq(IV 1) C = ssq(IV 2) B = ssq(Error) A+C+B = ssq(Total) Now we can think of the effect of IV 1 as A compared to either: A B IV 1 B+C DV or just B C IV 2

PARTIAL &SEMIPARTIAL • If we give the effect size IV 1 as A/(A+B+C), ‣ how much of the total variance of DV is explained by IV 1 ‣ semi-partial effect • If we give the effect size IV 1 as A/(A+B) ‣ how much of the remaining variance of DV after taking IV 2 into account ‣ partial effect

2 IVS - COVARIATION • Each circle represents the variance of that variable • Where they overlap, there is an effect IV 1 DV IV 1 has strong relationship with DV IV 2 has weak relationship with DV IV 1 has strong relationship with IV 2

2 IVS COVARIATION A = ssq(IV 1) C = ssq(IV 2) B = ssq(Error) A+C+B+D = ssq(Total) A IV 1 effects: Total Effect = (A+D)/(A+B+C+D) IV 1 DV Semipartial Effect = A/(A+B+C+D) Partial Effect = A/(A+B) B IV 2 C D

PARTIAL &SEMIPARTIAL • If we give the effect size IV 1 as A/(A+B+C+D), ‣ how much of the total variance of DV is explained uniquely by IV 1 ‣ = semi-partial effect • If we give the effect size IV 1 as A/(A+B) ‣ how much of the remaining variance of DV after taking IV 2 into account ‣ = partial effect

PARTIAL &SEMIPARTIAL • Semi-partial effect: A ‣ unique contribution of IV 1 to DV • Partial effect: ‣ contribution of IV 1 to DV after other variables are taken into account ‣ net gain in adding IV 1 to predictors B IV 1 DV IV 2 ‣ “value” of IV 1 C D

DIRECT EFFECTS • If we add all total effects, then ‣ area D gets counted multiple times. • If we add all semi-partial effects, then ‣ area D gets left out • Adding all partial effects doesn’t make sense ‣ they all have different denominators • Direct effects share out D amongst the IVs