Presentation on theme: "Overview of Lecture Factorial Designs Experimental Design Names"— Presentation transcript:
1Overview of LectureFactorial DesignsExperimental Design NamesPartitioning the VariablilityThe Two-Way Between Groups ANOVAEvaluating the Null HypothesesMain effectsInteractionsAnalytical Comparisons
2Factorial DesignMuch experimental psychology asks the question:What effect does a single independent variable have on a single dependent variable?It is quite reasonable to ask the following question as well.What effects do multiple independent variables have on a single dependent variable?Designs which include multiple independent variables are known as factorial designs.
3An example factorial design If we were looking at GENDER and TIME OF EXAM, these would be two independent factorsGENDER would only have two levels: male or femaleTIME OF EXAM might have multiple levels, e.g. morning, noon or nightThis is a factorial design
4Experimental Design Names The name of an experimental design depends on three pieces of informationThe number of independent variablesThe number of levels of each independent variableThe kind of independent variableBetween GroupsWithin Subjects (or Repeated Measures)
5Experimental design names If there is only one independent variable thenThe design is a one-way design (e.g. does coffee drinking influence exam scores)If there are two independent variablesThe design is a two-way design (e.g. does time of day or coffee drinking influence exam scores).If there are three independent variablesThe design is a three-way design (e.g. does time of day, coffee drinking or age influence exam scores).
6Experimental Design Names If there are 2 levels of the first IV and 3 levels of the second IVIt is a 2x3 designE.G.: coffee drinking x time of dayFactor coffee has two levels: cup of coffee or cup of waterFactor time of day has three levels: morning, noon and nightIf there are 3 levels of the first IV, 2 levels of the second IV and 4 levels of the third IVIt is a 3x2x4 designE.G.: coffee drinking x time of day x exam durationFactor coffee has three levels: 1 cup, 2 cup 3 cupsFactor time of day has two levels: morning or nightFactor exam duration has 4 levels: 30min, 60min, 90min, 120min
7Experimental Design Names If all the IVs are between groups thenIt is a Between Groups designIf all the IVs are repeated measuresIt is a Repeated Measures designIf at least one IV is between groups and at least one IV is repeated measuresIt is a Mixed or Split-Plot design
8Experimental design names Three IVsIV 1 is between groups and has two levels (e.g. a.m., p.m.)IV 2 is between groups and has two levels (e.g. coffee, water).IV 3 is repeated measures and has 3 levels (e.g. 1st year, 2nd year and 3rd year).The design is:A three-way (2x2x3) mixed design.
9Experimental design names The effect of a single variable is known as a main effectThe effect of two variables considered together is known as an interactionFor the two-way between groups design, an F-ratio is calculated for each of the following:The main effect of the first variableThe main effect of the second variableThe interaction between the first and second variables
10Analysis of a 2-way between groups design using ANOVA To analyse the two-way between groups design we have to follow the same steps as the one-way between groups designState the Null HypothesesPartition the VariabilityCalculate the Mean SquaresCalculate the F-Ratios
11Null HypothesesThere are 3 null hypotheses for the two-way (between groups design.The means of the different levels of the first IV will be the same, e.g.The means of the different levels of the second IV will be the same, e.g.The differences between the means of the different levels of the interaction are not the same, e.g.
12An example null hypothesis for an interaction The differences betweens the levels of factor A are not the same.
13Partitioning the variability If we consider the different levels of a one-way ANOVA then we can look at the deviations due to the between groups variability and the within groups variability.If we substitute AB into the above equation we getThis provides the deviations associated with between and within groups variability for the two-way between groups design.
14Partitioning the variability The between groups deviation can be thought of as a deviation that is comprised of three effects.In other words the between groups variability is due to the effect of the first independent variable A, the effect of the second variable B, and the interaction between the two variables AxB.
15Partitioning the variability The effect of A is given bySimilarly the effect of B is given byThe effect of the interaction AxB equalswhich is known as a residual
16The sum of squaresThe sums of squares associated with the two-way between groups design follows the same form as the one-wayWe need to calculate a sum of squares associated with the main effect of A, a sum of squares associated with the main effect of B, a sum of squares associated with the effect of the interaction.From these we can estimate the variability due to the two variables and the interaction and an independent estimate of the variability due to the error.
17The mean squaresIn order to calculate F-Ratios we must calculate an Mean Square associated withThe Main Effect of the first IVThe Main Effect of the second IVThe Interaction.The Error Term
18The mean squaresThe main effect mean squares are given by:The interaction mean squares is given by:The error mean square is given by:
19The F-ratiosThe F-ratio for the first main effect is:The F-ratio for the second main effect is:The F-ratio for the interaction is:
20An example 2x2 between groups ANOVA Factor A - Lectures (2 levels: yes, no)Factor B - Worksheets (2 levels: yes, no)Dependent Variable - Exam performance (0…30)MeanStd ErrorLECTURESWORKSHEETSyes19.2002.04no25.0001.2316.0001.709.6000.81
21Results of ANOVAWhen an analysis of variance is conducted on the data (using Experstat) the following results are obtainedSourceSum of SquaresdfMean SquaresFpA (Lectures)137.6040.000B (Worksheets)0.4500.0390.846AB16.1780.001Error1611.500
22What does it mean? - Main effects A significant main effect of Factor A (lectures)“There was a significant main effect of lectures (F1,16=37.604, MSe=11.500, p<0.001). The students who attended lectures on average scored higher (mean=22.100) than those who did not (mean=12.800).No significant main effect of Factor B (worksheets)“The main effect of worksheets was not significant (F1,16=0.039, MSe=11.500, p=0.846)”
23What does it mean? - Interaction A significant interaction effect“There was a significant interaction between the lecture and worksheet factors (F1,16=16.178, MSe=11.500, p=0.001)”However, we cannot at this point say anything specific about the differences between the means unless we look at the null hypothesisMany researches prefer to continue to make more specific observations.MeanStd ErrorLECTURESWORKSHEETSyes19.2002.04no25.0001.2316.0001.709.6000.81
24Simple main effects analysis We can think of a two-way between groups analysis of variance as a combination of smaller one-way anovas.The analysis of simple main effects partitions the overall experiment in this wayWorksheetsYes NoLectures(Yes)WorksheetsWorksheetsYes NoYesNoLectures(No)YesLecturesNoWorksheets (yes)Worksheets (no)YesYesLecturesLecturesNoNo
25Results of a simple main effects analysis Using ExperStat it possible to conduct a simple main effects analysis relatively easilySource of Sum of df Mean F pVariation Squares SquaresLectures atworksheets(yes)worksheets(no)Error TermWorksheets atlectures(yes)lectures(no)
26What does it mean? - Simple main effects of Lectures No significant simple main effect of lectures at worksheets (yes)“There was no significant difference between those students who did attend lectures (mean=19.20) or did not attend lectures (mean=16.00) when they completed worksheets (F1,16=2.226, MSe=11.500, p=0.155).”Significant simple main effect of lectures at worksheets (no)“There was a significant difference between those students who did attend lectures (mean=25.00) or did not attend lectures (mean=9.60) when they did not complete worksheets (F1,16=51.557, MSe=11.500, p<0.001). When students who attended lectures did not complete worksheets they scored higher on the exam than those students who neither attended lectures nor completed worksheets.”
27What does it mean? - Simple main effects of worksheets Significant simple main effect of worksheets at lectures (yes)“There was a significant difference between those students who did complete worksheets (mean=19.20) or did not complete worksheets (mean=25.00) when they attended lectures (F1,16=7.313, MSe=11.500, p=0.016). Students who attended the lectures and completed worksheets did less well than those students who attended lectures but did not complete the worksheets.”Significant simple main effect of worksheets at lectures (no)“There was a significant difference between those students who did complete worksheets (mean=16.00) or did not complete worksheets (mean=9.60) when they did not attend lectures (F1,16=51.557, MSe=11.500, p<0.001). When students who attended lectures did not complete worksheets they scored higher on the exam than those students who neither attended lectures nor completed worksheets.”
28Analytic comparisons in general If there are more than two levels of a FactorAnd, if there is a significant effect (either main effect or simple main effect)Analytical comparisons are required.Post hoc comparisons include tukey tests, Scheffé test or t-tests (bonferroni corrected).