Factorial Research Hypotheses & kxk BG Factorial Designs

Factorial Research Hypotheses & kxk BG Factorial Designs • Factorial RH – 3 types • Identifying Simple Effects to Directly Test the RH: • expanding the 2x2 design • reasons for larger designs • statistical analysis of kxk BG factorial designs • using LSD for kxk factorial designs

RH: for Factorial Designs • Research hypotheses for factorial designs may include • RH: for main effects • the effects of one IV, while ignoring the other IV • tested by comparing the appropriate marginal means • RH for Simple Effects • the effect of one IV as a particular value of the other IV • tested by comparing the cell means of the appropriate cells • RH: for interactions • usually expressed as “different differences” -- differences between a set of simple effects • tested by comparing the results of the appropriate set of simple effects • That’s the hard part -- determining which set of simple effects gives the most direct test of the interaction RH:

Is each an Interaction or a Main Effects RH:??? (Hint – read them in pairs!) Males tend to outperform females on standardized math tests. Males tend to outperform females on standardized math tests, however the difference decreases with age. ME INT Improvement during therapy for depression occurs faster for those receiving cognitive-behavioral therapy than for those receiving traditional psychodynamic therapy. Therapy for depression is generally effective. INT ME Young girls and boys have similar overall skill levels. Young girls have better verbal skills than motor skills, however young boys have better motor skills than verbal skills. ME INT

Sometimes the Interaction RH: is explicitly stated • when that happens, one set of SEs will provide a direct test of the RH: (the other won’t) Presentation Comp Paper Here’s an example: Easy tasks will be performed equally well using paper or computer presentation, however, hard tasks will be performed better using computer presentation than paper. Task Diff. Easy Hard = > This is most directly tested by inspecting the simple effect of paper vs. computer presentation for easy tasks, and comparing it to the simple effect of paper vs. computer for hard tasks.

Your Turn... Young boys will rate playing with an electronic toy higher than playing with a puzzle, whereas young girls will have no difference in ratings given to the two types of toys. Type of Toy Elec. Puzzle Gender Boys Girls > = Type of Evidence Confession Witness Who Judge Lawyer Judges will rate confessions as more useful than eyewitness testimony, whereas Lawyers will rate eyewitness testimony as more useful than confessions. > <

Sometimes the set of SEs to examine use is “inferred” ... • Often one of the IVs in the study was used in previous research, and the other is “new”. • In this case, we will usually examine the simple effect of the “old” variable, at each level of the “new” variable • this approach gives us a clear picture of the replication and generalization of the “old” IV’s effect. e.g., Previously I demonstrated that computer presentations lead to better learning of statistical designs than does using a conventional lecture. I would like to know if the same is true for teaching writing. Let’s take this “apart” to determine which set of SEs to use to examine the pattern of the interaction...

Previously I demonstrated that computer presentations lead to better learning of statistical designs than does using a conventional lecture. I would like to know if the same is true for teaching writing. Type of Instruction Comp Lecture Here’s the design and result of the earlier study about learning stats. > Here’s the design of the study being planned. Type of Instruction Comp Lecture Topic Stats Writing What cells are a replication of the earlier study ? So, which set of SEs will allow us to check if we got the replication, and then go on to see of we get the same results with the new topic ? Yep, SE of Type of Instruction, for each Topic ...

Take a look at these .. #1 I have previously demonstrated that rats learn Y-mazes faster than to hamsters. I wonder if the same is true for radial mazes ? Would look at the SE of ________________________ & SE of ________________________ #2 I’ve discovered that Psyc and Soc majors learn statistics about equally well. My next research project will also compare these types of students on how well they learn research ethics. Would look at the SE of ________________________ & SE of ________________________

Sometimes the RH: about the interaction and one of the main effects are “combined” • this is particularly likely when the expected interaction pattern is of the > vs. >type • Here’s an example… Type of Therapy Group Indiv. Anxiety Social Agora. > Group therapy tends to work better than individual therapy,although this effect is larger for patients with social anxiety than with agoraphobia. > > Int. RH: Main effect RH: So, we would examine the interaction by looking at the SEs of Type of Therapy for each type of Anxiety.

About the causal interpretation of effects of a factorial design… Start by assessing the causal interpretability of each main effect In order to causally interpret an interaction, you must be able to casually interpret BOTH main effects. Study of Age and Gender no casually interpretable effects (main effects nor interaction) Study of Age and Type of Toy (RA + Manip) only casually interpretable effect would be the main effect of Type of Toy (not the main effect of Age, nor the interaction). Study Type of Toy (RA + Manip) and Playing Situation (RA + manip) all effects are causally interpreted (both main effects and the interaction).

Basic and Expanded Factorial Designs The simplest factorial design is a 2x2, which can be expanded in two ways: 1) Adding conditions to one, the other, or both IVs 2x4 design 2x2 design 3x2 design 3x4 design

Factorial designs are all labeled as ? x ? Designs • the first number tells the number of rows in the design • the second number tells the number of columns in the design • What is each of the following 5 x 3 a. c. 2 x 3 b. 4 x 6

This time try to “draw the boxes” (be sure to label each IV and specify each or its conditions) based on the description of the design -- whatever IV is described first will define the rows 1. Males and females completed the task, either under instructions to work quickly, work accurately, to work as quickly as possible without making unnecessary errors or no instructions. 2. Folks completed a depression questionnaire either under instructions to “respond like someone with acute depression,” “respond like someone with chronic depression” or “respond like someone who is trying to ‘fake’ being depressed”. Participants were either clinical psychologists, clinical Ph.D., or volunteers from a local social club.

#1 was a 2x4 design that looks like this... Instructions Quick Accurate Both None Gender Male Female #2 was a 3 x 3 design that looks like ... Participant Clinician Clin. Grad. Soc. Club Respond like a .. Acute depressive Chronic depressive “Fake” depressive

kxk BG Factorial Designs • We’ve worked extensively with the 2x2 design -- the basic factorial • Larger factorial designs are often used for the same reasons that multiple-condition 1-factor designs are used . . . • You may need more than 2 IV conditions to properly test a RH: • Want multiple “experimental conditions” (qual or quant diffs) • Want multiple “treatment conditions “ (standard vs. none, etc) • Want to “dissect” a multiple element treatment • You might want to test the replicability of an IV’s effect across more than two situations/settings • testing the generality of a TX across gender requires just 2 conditions of the 2nd IV • testing generality of that TX across ages would require more

Statistical Analysis of kxk Factorial Designs • Only a couple of differences from the 2x2 • 1. Tell IVs and DV 2. Present data in table or figure • 3. Determine if the interaction is significant • if it is, describe it in terms of one of the sets of simple effects using LSD mmd to compare the cell means • 4. Determine whether or not the first main effect is significant • if so, describe it using LSD mmd compare 3+ marginal means • determine if that main effect is descriptive or misleading • 5. Determine whether or not the second main effect is significant • if so, describe it using LSD mmd compare 3+ marginal means • determine if that main effect is descriptive or misleading

The omnibus ANOVA for the kxk is the same as for the 2x2 • BG SStotal = SSA + SSB + SSINT + SSError • dftotal = dfA + dfB + dfINT + dfError • (N - 1) = ( a -1) + (b-1) + (a-1)(b-1) + ab(n-1) • SSA / dfA SSB / dfB SSINT / dfINT FA = --------------- FB = -------------- FINT = ----------------- SSE / dfE SSE / dfE SSE / dfE • Things to notice: • There is a single error term that is used for all the Fs • All of the effects are equally “powerful” except for sample size differences (stat power)

The LSD follow-ups are a little different than for the 2x2 • the 2x2 uses the LSD only for comparing cell means • describe the simple effects to explicate the interaction pattern • not needed for MEs , since they involve only 2 conditions • the kxk uses the LSD for comparing cell & marginal means • different LSD mmd values are computed for different effects • if the interaction is significant, then an LSD is computed to compare the cell means -- describe SEs, interaction, etc. • If a ME with 2 conditions is significant - no LSD needed • If a ME with 3 or more conditions is significant, then LSD is computed to compare the marginal means of that ME • Be sure to use the proper “n” to compute each LSD • “n” = mean number of data points used to compute the means being compared (more on demo sheet)

What statistic is used for which factorial effects???? Gender Male Female Age 5 10 15 30 30 30 20 30 25 25 30 27.5 • “Effects” in this sudy • Main effect of gender • Main effect of age • Pairwise age ME effects • Interaction of age & gender • Pairwise SE of age for males • Pairwise SE of age for females • SE of gender for 5 yr olds • SE of gender for 10 yr olds 25 30 27.5 • There will be 5 statistics • FGender • FAge • LSDmmd • FInt • LSDmmd

Instruction Quick Accurate Both None Back to  100 males and 100 females completed the task, either under instructions to work quickly, work accurately, to work as quickly as possible without making unnecessary errors or no instructions. Gender Male Female • For the interaction p = .03 • will we need and LSDmmd to explore the pattern of the interaction? why or why not? • what will “n” be? Yep - k = 8 ! 200 / 8 = 25 • For the main effect of instruction p = .02 • will we need and LSDmmd to explore the pattern of this main effect ? why or why not? • what will “n” be? Yep – k = 5 ! 200 / 4 = 50 • For the main effect of gender p = .02 • will we need and LSDmmd to explore the pattern of this main effect ? why or why not? • what will “n” be? Nope – k = 2 !

Factorial Research Hypotheses & kxk BG Factorial Designs