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Regression Models w/ 2 Categorical Variables Sources of data for this model Variations of this model Definition and advantage of factorial research designs.

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Presentation on theme: "Regression Models w/ 2 Categorical Variables Sources of data for this model Variations of this model Definition and advantage of factorial research designs."— Presentation transcript:

1 Regression Models w/ 2 Categorical Variables Sources of data for this model Variations of this model Definition and advantage of factorial research designs 5 terms necessary to understand factorial designs 5 patterns of factorial results for a 2x2 factorial designs

2 As always, “the model doesn’t care where the data come from”. Those data might be … 2 measured categorical variables (e.g., gender & species) 2 manipulated categorical variables (e.g., Tx1 v Tx2 & FB v nFB) a measured categorical variable (e.g., species) and a manipulated categorical variable (e.g., FB v nFB) Like nearly every model in the ANOVA/regression/GLM family – this model was developed for and originally applied to experimental designs with the intent of causal interpretability !!! As always, causal interpretability is a function of design (i.e., assignment, manipulation & control procedures) – not statistical model or the constructs involved !!!

3 There are two important variations of this model 1.Main effects model Terms for both categorical variables No interaction – assumes simple effect homgeneity b-weights for the 2 categorical variables each represent main effect of that variable 2. Interaction model Terms for both categorical variables Term for interaction - does not assume SE homogen !! b-weights for 2 qual variables each represent the simple effect of that variable when the other variable = 0 b-weight for the interaction term represented how the simple effect of one variable changes with changes in the value of the other variable (e.g., the extent and direction of the interaction)

4 Type of Therapy Initial Diagnosis GroupIndividual General clients diagnosed w/ clients diagnosed w/ Anxiety general anxiety who general anxiety who received group therapy received individual therapy Social clients diagnosed w/ clients diagnosed w/ Anxiety social anxiety who social anxiety who received group therapy received individual therapy Participants in each “cell” of this design have a unique combination of IV conditions. Showing this design is a 2x2 Factorial

5 The difficult part of learning about factorial designs is the large set of new terms that must be acquired. Here’s a summary;; cell means -- the mean DV score of all the folks with a particular combination of IV treatments marginal means -- the mean DV score of all the folks in a particular condition of the specified IV (aggregated across conditions of the other IV) Main effects involve the comparison of marginal means. Simple effects involve the comparison of cell means. Interactions involve the comparison of simple effects.

6 Identifying Cell Means and Marginal Means Type of Therapy Initial DiagnosisGroup Individual General Anxiety 50 5050 Social 90 1050 Anxiety 70 30 Cell means  mean DV of subjects in a design cell Marginal means  average mean DV of all subjects in one condition of an IV

7 Identifying Main Effects -- difference between the marginal means of that IV (ignoring the other IV) Type of Therapy Initial DiagnosisGroup Individual General Anxiety 50 5050 Social Anxiety 90 1050 70 30 Main effect of Initial Diagnosis Main effect of Type of Therapy

8 Identifying Simple Effects -- cell means differences between conditions of one IV for a specific level of the other IV Type of Therapy Initial DiagnosisGroupIndividual General Anxiety 50 501 Social Anxiety 90 102 ab Simple effects of Initial Diagnosis for each Type of Therapy aSimple effect of Initial Diagnosis for group therapy bSimple effect of Initial Diagnosis for individual therapy

9 Identifying Simple Effects -- cell means differences between conditions of one IV for a specific level of the other IV Type of Therapy Initial DiagnosisGroupIndividual General Anxiety 50 501 Social Anxiety 90 102 a b Simple effects of Type of Therapy for each Initial Diagnosis 1 Simple effect of Type of Therapy for general anxiety patients 2 Simple effect of Type of Therapy for social anxiety patients

10 5 basic factorial data patterns - 3 have interactions – 2 do not #1 Task Presentation Paper Computer Task Difficulty Easy 90 > 70 simple effects are Hard 40 < 60 opposite directions There is an interaction of Task Presentation and Task Difficulty as they relate to performance. Easy tasks are performed better using paper than using computer (90 vs. 70), whereas hard tasks are performed better using the computer than using paper (60 vs. 40).

11 #1a – an important variation of this data pattern I call this, “The effect that isn’t anywhere!” There is a significant interaction(p <.05). But – neither simple effect is statistically significant !?!?! Task Presentation Paper Computer Task Difficulty Easy 90 = 70. Hard 40 = 60 simple effects look like they’re in opposite directions, but are both nulls!!! Remember! An interaction is that the SEs are “different from each other”. The significant interaction tells us the 90 > 70 simple effect is different from the 40 < 60 simple effect.

12 #2 Task Presentation Paper Computer Task Difficulty Easy 90 = 90 one simple effect “null” Hard 40 < 70 one simple effect There is an interaction of Task Presentation and Task Difficulty as they relate to performance. Easy tasks are performed equally well using paper and using the computer (90 vs. 90), however, hard tasks are performed better using the computer than using paper (70 vs. 40).

13 #3 Task Presentation Paper Computer Task Difficulty Easy80 < 90 simple effects in the same direction, Hard 40 < 70 but of different sizes There is an interaction of Task Presentation and Task Difficulty as they relate to performance. Performance was better using the computer than using paper, however this effect was larger for hard tasks (70 vs. 40) than for easy tasks (90 vs. 80).

14 There are the two basic patterns of NON-interactions #4 Task Presentation Paper Computer Task Difficulty Easy 30 < 50 both simple effects are in the same direction and are Hard 50 < 70 the same size There is no interaction of Task Presentation and Task Difficulty as they relate to performance. Performance is better for computer than for paper presentations (for both Easy and Hard tasks). Notice the main effects will be descriptive.

15 #5 Task Presentation Paper Computer Task Difficulty Easy 50 = 50 both simple effects Hard 70 = 70 are nulls There is no interaction of Task Presentation and Task Difficulty as they relate to performance. Performance is the same for computer and paper presentations (for both Easy and Hard tasks). Notice the main effects will be descriptive.

16 1 simple effects in opposite directions 2. = vs. < one null simple effect and one simple effect 3. < vs. < simple effects in same direction, but different sizes 4. < vs. < simple effects of the same size in the same direction 5. = vs. = both null simple effects We can rearrange the 5 basic patterns of results from a 2x2 Factorial to help us think about interactions and descriptive/misleading main effects Interaction -- simple effects of different size and/or direction Misleading main effects Descriptive main effects No Interaction -- simple effects are null or same size

17 Here’s yet another way of thinking about interactions… we’ll need an example!! Brownies – great things… worthy of serious theory & research!!! The usual brownie is made with 4 blocks of chocolate and 2 cups of sugar. Replicated research tells us that the average rating of brownies made with this recipe is about 3 on a 10- point scale. My theory? People don’t really like brownies! What they really like is fudge! So, goes my theory, making brownies more “fudge-like” will make them better liked. How to make them more fudge-like, you ask? Add more sugar & more chocolate!!!

18 So, we made up several batches of brownies and asked people to taste a standardized amount of brownie after rinsing their mouth with water, eating an unsalted saltine cracker and rinsing their mouth a second time. We used the same 10-point rating scale; 1 = this is the worst plain brownie I’ve ever had, 10=this is the best plain brownie I’ve ever had. Our first study: 2-cups of sugar 4-cups of sugar 35 So, far so good!

19 Our second study: 4 blocks of choc. 32 8 blocks of choc. What????Oh – yeah! Unsweetened chocolate… Then the argument started.. One side: We have partial support for the theory – adding sugar helps, but adding chocolate hurts!!! Other side: We have not tested the theory!!! What was our theory? Add more sugar & more chocolate!!! We need a better design!

20 4 blocks of choc. 32 8 blocks of choc. 2-cups of sugar 4-cups of sugar 5 What do we expect for the 4-cup & 8-block brownies? standard brownie + sugar effect + chocolate effect expected additive effect of choc & sugar expected score for 4&8 brownies 3 + 2 - 1 1 4

21 4 blocks of choc. 32 8 blocks of choc. 2-cups of sugar 4-cups of sugar 5 How do we account for this ? 9 There is a non-additive joint effect of chocolate and sugar!!!! The joint effect of adding chocolate and sugar is not predictable as the sum of the effects of adding each!!! Said differently, there is an interaction of chocolate and sugar that emerges when they are added simultaneously. The effect of adding both simultaneously is 6 … not 1???

22 4 blocks of choc. = 0 32 8 blocks of choc. = 1 2-cups of sugar = 0 4-cups of sugar = 1 59 DV’ = b1*sugar + b2 *choc + b3*Int + a a = DV’ when sugar & choc = 0  3 b1 = SE of sugar when choc = 0  2 b2 = SE of choc when sugar = 0  -1 b3 = SE dif  5  SE of sugar when choc = 1  2 + 5 = 7  SE of choc when sugar = 1  -1 + 5 = 4

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