Situations with more than two variables of interest

When considering the relationship among three or more variables, an interaction may arise. Interactions describe a situation in which the simultaneous influence of two variables on a third is not additive. Most commonly, interactions are considered in the context of Multiple Regression analyses, but they may also be evaluated using Two-Way ANOVA.

• Example Problem
• Two-Way ANOVA
• Multiple Regression

An Example Problem

Suppose we are interested in studying the effects of cocaine on sleep. We might design an experiment to simultaneously test whether both the use of cocaine and the duration of usage affect the number of hours a squirrel will sleep in a night. We might give half of the squirrels we test cocaine, and the other half a placebo substance (the substance variable). And we might vary the duration of usage by administering cocaine or placebo for one of two possible durations before test, 4 weeks or 12 weeks (the duration variable). We can then consider the average treatment response (e.g. number of hours slept) for each squirrel, as a function of the treatment combination that was administered (e.g. substance and duration). The following table shows one possible situation:

There are three null hypotheses we may want to test. The first two test the effects of each factor under investigation:

• H₀₁: Both substance groups sleep for the same number of hours on average.

• H₀₂: Both treatment duration groups sleep for the same number of hours on average.

And the third tests for an interaction between these two factors:

• H₀₃: The two factors are independent or there is no interaction effect.

Two-Way ANOVA

A two-way ANOVA is an analysis technique that quantifies how much of the variance in a sample can be accounted for by each of two categorical variables and their interactions.

Step 1 is to compute the group means (for each cell, row, and column):

Step 2 is to calculate the sum of squares (SS) for each group (cell) using the following formula:

where is the i'th measurement for group g, and is the overall group mean for group g.

For each group, this formula is implemented as follows:

• 4-Week Placebo Dataset = {7 8 6 7 6.5}, M = 7
  SS_{4-Wk Placebo} = (7-7)² + (8-7)² + (6-7)² + (7-7)² + (6.5-7)² = 2.25
  
  

  4-Week Cocaine = {5.5 3.5 4.5 6 5},

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  = 4.9
  SS_{4-Wk Cocaine} = (5.5-4.9)² + (3.5-4.9)² + (4.5-4.9)² + (6-4.9)² + (5-4.9)² =$)>> 3.7
  
  

  12-Week Placebo = {8 10 13 9 8.5}, M = 9.7
  SS_{12-Wk Placebo} = (8-9.7)² + (10-9.7)² + (13-9.7)² + (9-9.7)² + (8.5-9.7)² = 15.8
  
  

  12-Week Cocaine = {5 4.5 4 6 4}, M = 4.7
  SS_{12-Wk Cocaine} = (5-4.7)² + (4.5-4.7)² + (4-4.7)² + (6-4.7)² + (4-4.7)² = 2.8
  
  

Step 3 is to calculate the between-groups sum of squares(SS_B):

where n is the number of subjects in each group, is the mean for group g, and is the overall mean (across groups).

• SS_B = n [( M_{4-Wk Placebo} - M_Group )² + ( M_{4-Wk Cocaine} - M_Group)² + ( M_{12-Wk Placebo} - M_Group )² + ( M_{12-Wk Cocaine} - M_Group)²]
  

  • = 5 [(7 - 6.575 )² + (4.9 - 6.575)² + (9.7 - 6.575)² + (4.7 - 6.575)²]
    = 5 [0.180625 + 2.805625 + 9.765625 + 3.515625]
    = 5 [16.2675]
    = 81.3375
    
    

Now, Step 4 , we'll calculate the SS_W:

• SS_W = SS_{4-Wk Placebo} + SS_{4-Wk Cocaine} + SS_{12-Wk Placebo} + SS_{12-Wk Cocaine}
  

  • = 2.25 + 3.7 + 15.8 + 2.8
    = 24.55
    
    

  df_W = N - rc
  

  • = 20 - (2 * 2)
    = 16
    
    

  s_W² = SS_W / df_W
  

  • = 24.55 / 16
    = 1.534375
    
    

For Step 5, we'll calculate the SS_R:

• SS_R = n [( M_Placebo - M_Group )² + ( M _Cocaine - M_Group)²]
  

  • = 10 [(8.35 - 6.575)² + (4.8 - 6.575)²]
    = 10 [3.150625 + 3.150625]
    = 10 [6.30125]
    = 63.0125
    
    

  • df_R = r - 1
    

    • = 2-1
      = 1
      
      

    s_R² = SS_R / df_R
    

    • = 63.0125 / 1
      = 63.0125
      
      

In Step 6, we calculate the SS_C:

• SS_C = n [( M_4-Week - M_Group )² + ( M _12-Week - M_Group)²]
  

  • = 10 [(5.95 - 6.575)² + (7.2 - 6.575)²]
    = 10 [0.390625 + 0.390625]
    = 10 [0.78125]
    = 7.8125
    
    

  • df_C = c - 1
    

    • = 2-1
      = 1
      
      

    s_C² = SS_C / df_C
    

    • = 7.8125 / 1
      = 7.8125
      
      

In Step 7, calculate the SS_RC:

• SS_RC = SS_B - SS_R - SS_C
  

  • = 81.3375 - 63.0125 - 7.8125
    = 10.5125
    
    

  • df_RC = (r - 1)(c - 1)
    

    • = (2-1)(2-1)
      = 1
      
      

    s_RC² = SS_RC / df_RC
    

    • = 10.5125 / 1
      = 10.5125
      
      

Now, in Step 8, we'll calculate the SS_T:

• SS_T = SS_B + SS_W + SS_R + SS_C + SS_RC
  

  • = 81.3375+ 24.55 + 63.0125 + 7.8125 + 10.5125
    = 187.225
    
    

  df_T = N - 1
  

  • = 20-1
    = 19
    
    

At Step 9, we calculate the F values:

• F_R = s_R² / s_W²
  

  • = 63.0125 / 1.534375
    = 41.06721
    
    

  F_C = s_C² / s_W²
  

  • = 7.8125 / 1.534375
    = 5.09165
    
    

  F_RC = s_RC² / s_W²
  

  • = 10.5125 / 1.534375
    = 6.851324
    
    

And, finally, at Step 10, we can organize all of the above into a table, along with the appropriate F_CRIT values (looked up in a table like this one) that we'll use for comparison and interpretation of our computations:

• F_CRIT (1, 16) _α=0.5 = 4.49

[INTERPRETATION HERE]

Multiple Regression

[BASIC INFO ABOUT MULTIPLE REGRESSION]