Differences between revisions 122 and 123

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.

An Example Problem

To Measures of spine density were taken from both control (Cntl) and experimental (Coc: prenatal cocaine exposure) rats at 4 and 12 weeks of age.

Suppose we want to determine if prenatal exposure to cocaine alters dendritic spine density in prefrontal cortex over the course of development. We might design an experiment to simultaneously test whether prenatal exposure to cocaine and age affects the dendritic spine density in rats of two different ages. Suppose we took 20 rats, half of which were exposed to cocaine prenatally (the cocaine group) and half of which were not (the control group). We'll refer to the difference between these two groups (cocaine vs. control) as the prenatal-exposure variable. Now further suppose that each prenatal-exposure group is composed of an equal number of rats of two different ages: either 4-weeks or 12-weeks. We can refer to this difference as the age variable. We can then consider the average response of our dependent variable (e.g. dendritic spine density) for each rat, as a function of both variables (e.g. prenatal-exposure and age). The following table shows one possible outcome of such a study:

DENDRITIC SPINE DENSITY
4-Week Control	4-Week Cocaine	12-Week Control	12-Week Cocaine
7.5	5.5	8.0	5.0
8.0	3.5	10.0	4.5
6.0	4.5	13.0	4.0
7.0	6.0	9.0	6.0
6.5	5.0	8.5	4.0

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

H₀₁: Both prenatal-exposure groups have the same dendritic spine density on average.
H₀₂: Both age groups have the same dendritic spine density on average.

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

H₀₃: The two factors (pre-natal exposure and age) 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):

GROUP MEANS
	4-Week	12-Week	All Ages
Control	7	9.7	8.35
Cocaine	4.9	4.7	4.8
All Prenatal Exposures	5.95	7.2	6.575

It's important to plot the numbers--it's easier to understand them that way:

Means by group

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

$\sum_{i=1} (x_{i,g} - \overline{X}_g)^2$

where $x_{i,g}$ is the i'th measurement for group g, and $\overline{X}_g$ is the overall group mean for group g.

For each group, this formula is implemented as follows:

4-Week Control:
- {7.5, 8, 6, 7, 6.5}, $\overline{X}_{4-week, control}$ = 7
  $SS_{4-week, control}$ = (7.5-7)² + (8-7)² + (6-7)² + (7-7)² + (6.5-7)² = 2.5
4-Week Cocaine:
- {5.5, 3.5, 4.5, 6, 5}, $\overline{X}_{4-week, cocaine}$ = 4.9
  $SS_{4-week, cocaine}$ = (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 Control:
- {8, 10, 13, 9, 8.5}, $\overline{X}_{12-week, control}$ = 9.7
  $SS_{12-week, control}$ = (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}, $\overline{X}_{12-week, cocaine}$ = 4.7
  $SS_{12-week, cocaine}$ = (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):

$n \cdot \sum_{g} (\overline{X}_{g} - \overline{X})^2$

where n is the number of subjects in each group, $\overline{X}_g$ is the mean for group g, and $\overline{X}$ is the overall mean (across groups).

$SS_{B}$ = [( $\overline{X}_{4-week, control}$ - $\overline{X}$ )² + ( $\overline{X}_{4-week, cocaine}$ - $\overline{X}$ )² + ( $\overline{X}_{12-week, control}$ - $\overline{X}$ )² + ( $\overline{X}_{12-week, cocaine}$ - $\overline{X}$ )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 sum-of-squares within groups ( $SS_{W}$ ). For a group g, this is

$\sum_{g} SS_g$

So:

$SS_{W}$ = $SS_{4-week, control}$ + $SS_{4-week, cocaine}$ + $SS_{12-week, control}$ + $SS_{12-week, cocaine}$
- = 2.5 + 3.7 + 15.8 + 2.8
  = 24.8
$df_{w}$ =
- = 20 - (2 * 2)
  = 16
$s_{W}$ ² = $SS_{W}$ / $df_{w}$
- = 24.8 / 16
  = 1.55

For Step 5, we'll calculate the sum-of-squares for the rows ( $SS_{R}$ ):

$\sum_{r} (\overline{X}_r - \overline{X})^2$

where r ranges over rows.

$SS_{R}$ = [( $\overline{X}_{control}$ - $\overline{X}$ )² + ( $\overline{X}_{cocaine}$ - $\overline{X}$ )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 sum-of-squares for the columns ( $SS_{C}$ ):

$\sum_{c} (\overline{X}_c - \overline{X})^2$

where c ranges over columns.

$SS_{C}$ = ( $\overline{X}_{4-week}$ - $\overline{X}$ )² + ( $\overline{X}_{12-week}$ - $\overline{X}$ )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 sum-of-squares for the interaction ( $SS_{RC}$ ):

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 total sum-of-squares ( $SS_{T}$ ):

SS_T = SS_B + SS_W + SS_R + SS_C + SS_RC
- = 81.3375 + 24.8 + 63.0125 + 7.8125 + 10.5125
  = 187.475
df_T = N - 1
- = 20-1
  = 19

At Step 9, we calculate the F values:

F_R = s_R² / s_W²
- = 63.0125 / 1.55
  = 40.65323
F_C = s_C² / s_W²
- = 7.8125 / 1.55
  = 5.040323
F_RC = s_RC² / s_W²
- = 10.5125 / 1.55
  = 6.782258

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

ANOVA TABLE
Source	SS	df	s²	F_obt	F_crit	p
rows	63.0125	1	63.0125	40.65323	4.49	p < 0.05
columns	7.8125	1	7.8125	5.040323	4.49	p < 0.05
r * c	10.5125	1	10.5125	6.782258	4.49	p < 0.05
within	24.8	16	1.55	--	--	--
total	187.475	19	--	--	--	--

Both factors (prenatal-exposure and age) are significant, as indicate by the fact that F_OBT > F_CRIT. Thus, we can reject all H₀₁ and H₀₂ and conclude that dendritic spine density is affected by prenatal exposure and age. The interaction between the two factors (r * c) is also significant. Thus, we can also reject H₀₃ and conclude there is a significant interaction between these two factors.

To further interpret these results, we can plot the group means as follows:

Means by group

So, as indicated above, spine density is significantly greater in the control group as compared to drug-exposed animals at both 4 and 12 weeks of age. The effect of prenatal drug exposure is magnified with time due to a developmental increase in spine density that occurs only in the control group. Between 4 and 12 weeks of age, spine density in prefrontal cortex increases significantly in control animals, while it remains relatively unchanged in the drug-exposed animals.

Multiple Regression

Another analysis technique you could use is a multiple regression. In multiple regression, we find coefficients for each group such that we are able to best predict the group means. The multiple regression computes standard errors on the coefficients, meaning that we can determine if a coefficient is significantly different from zero. On this simple example, multiple regression will give identical answers to ANOVA, but in more complex cases, multiple regression is a more powerful technique that allows you to include additional nuisance predictors, that the analysis controls for before testing for significance of your independent variables.

-  ⇤ ← Revision 122 as of 2011-11-30 02:36:50 → 
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  Editor: CelesteKidd
  Comment:
+   ← Revision 123 as of 2011-11-30 03:40:19 → ⇥
  Size: 11630
  Editor: CelesteKidd
  Comment:
-Deletions are marked like this.
+Additions are marked like this.
 Line 10:
-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:

||||||||HOURS SLEPT IN SINGLE NIGHT||
|| '''4-Week Placebo (Control)''' || '''4-Week Cocaine''' || '''12-Week Placebo (Control)''' || '''12-Week Cocaine''' ||
+To   Measures of spine density
were taken from both control (Cntl) and experimental (Coc: prenatal
cocaine exposure) rats at 4 and 12 weeks of age. 


Suppose we want to determine if prenatal exposure to cocaine alters dendritic spine density in prefrontal cortex over the course of development. We might design an experiment to simultaneously test whether prenatal exposure to cocaine and age affects the dendritic spine density in rats of two different ages. Suppose we took 20 rats, half of which were exposed to cocaine prenatally (the ''cocaine'' group) and half of which were not (the ''control'' group). We'll refer to the difference between these two groups (''cocaine'' vs. ''control'') as the ''prenatal-exposure'' variable. Now further suppose that each prenatal-exposure group is composed of an equal number of rats of two different ages: either 4-weeks or 12-weeks. We can refer to this difference as the ''age'' variable. We can then consider the average response of our dependent variable (e.g. dendritic spine density) for each rat, as a function of both variables (e.g. prenatal-exposure and age). The following table shows one possible outcome of such a study:

||||||||DENDRITIC SPINE DENSITY||
|| '''4-Week Control''' || '''4-Week Cocaine''' || '''12-Week Control''' || '''12-Week Cocaine''' ||
-Line 20:
+Line 26:
-There are '''three null hypotheses''' we may want to test. The first two test the effects of each factor under investigation:

 * H,,01,,: Both substance groups sleep for the same number of hours on average.
 * H,,02,,: Both treatment duration groups sleep for the same number of hours on average.
+There are '''three null hypotheses''' we may want to test. The first two test the effects of each variable (or factor) under investigation:

 * H,,01,,: Both prenatal-exposure groups have the same dendritic spine density on average.
 * H,,02,,: Both age groups have the same dendritic spine density on average.
-Line 27:
+Line 33:
- * H,,03,,: The two factors are independent or there is no interaction effect.
+ * H,,03,,: The two factors (pre-natal exposure and age) are independent or there is no interaction effect.
-Line 36:
+Line 42:
-||   || '''4-Week''' || '''12-Week''' || '''''All Durations''''' ||
|| '''Placebo''' || 7 || 9.7|| ''8.35'' ||
+||   || '''4-Week''' || '''12-Week''' || '''''All Ages''''' ||
|| '''Control''' || 7 || 9.7|| ''8.35'' ||
-Line 39:
+Line 45:
-|| '''''All Substances''''' || ''5.95'' || ''7.2'' || ''6.575'' ||
+|| '''''All Prenatal Exposures''''' || ''5.95'' || ''7.2'' || ''6.575'' ||
-Line 57:
+Line 63:
--Week Placebo: <<BR>>
     {7.5, 8, 6, 7, 6.5}, <<latex($\overline{X}_{4-week, placebo}$)>> = 7<<BR>>
     <<latex($SS_{4-week, placebo}$)>> = (7.5-7)^2^ + (8-7)^2^ + (6-7)^2^ + (7-7)^2^ + (6.5-7)^2^ = '''2.5'''<<BR>><<BR>>
+-Week Control: <<BR>>
     {7.5, 8, 6, 7, 6.5}, <<latex($\overline{X}_{4-week, control}$)>> = 7<<BR>>
     <<latex($SS_{4-week, control}$)>> = (7.5-7)^2^ + (8-7)^2^ + (6-7)^2^ + (7-7)^2^ + (6.5-7)^2^ = '''2.5'''<<BR>><<BR>>
-Line 65:
+Line 71:
--Week Placebo: <<BR>>
     {8, 10, 13, 9, 8.5}, <<latex($\overline{X}_{12-week, placebo}$)>> = 9.7<<BR>>
     <<latex($SS_{12-week, placebo}$)>> = (8-9.7)^2^ + (10-9.7)^2^ + (13-9.7)^2^ + (9-9.7)^2^ + (8.5-9.7)^2^ = '''15.8'''<<BR>><<BR>>
+-Week Control: <<BR>>
     {8, 10, 13, 9, 8.5}, <<latex($\overline{X}_{12-week, control}$)>> = 9.7<<BR>>
     <<latex($SS_{12-week, control}$)>> = (8-9.7)^2^ + (10-9.7)^2^ + (13-9.7)^2^ + (9-9.7)^2^ + (8.5-9.7)^2^ = '''15.8'''<<BR>><<BR>>
-Line 85:
+Line 91:
-  <<latex($SS_{B}$)>> =  <<latex($n$)>> [( <<latex($\overline{X}_{4-week, placebo}$)>> - <<latex($\overline{X}$)>> )^2^ + ( <<latex($\overline{X}_{4-week, cocaine}$)>> - <<latex($\overline{X}$)>>)^2^ + ( <<latex($\overline{X}_{12-week, placebo}$)>> - <<latex($\overline{X}$)>> )^2^ + ( <<latex($\overline{X}_{12-week, cocaine}$)>> - <<latex($\overline{X}$)>>)^2^]<<BR>>
+  <<latex($SS_{B}$)>> =  <<latex($n$)>> [( <<latex($\overline{X}_{4-week, control}$)>> - <<latex($\overline{X}$)>> )^2^ + ( <<latex($\overline{X}_{4-week, cocaine}$)>> - <<latex($\overline{X}$)>>)^2^ + ( <<latex($\overline{X}_{12-week, control}$)>> - <<latex($\overline{X}$)>> )^2^ + ( <<latex($\overline{X}_{12-week, cocaine}$)>> - <<latex($\overline{X}$)>>)^2^]<<BR>>
-Line 101:
+Line 107:
-   <<latex($SS_{W}$)>> = <<latex($SS_{4-week, placebo}$)>> + <<latex($SS_{4-week, cocaine}$)>> + <<latex($SS_{12-week, placebo}$)>> + <<latex($SS_{12-week, cocaine}$)>><<BR>>
+   <<latex($SS_{W}$)>> = <<latex($SS_{4-week, control}$)>> + <<latex($SS_{4-week, cocaine}$)>> + <<latex($SS_{12-week, control}$)>> + <<latex($SS_{12-week, cocaine}$)>><<BR>>
-Line 123:
+Line 129:
-  <<latex($SS_{R}$)>> =  <<latex($n$)>> [( <<latex($\overline{X}_{placebo}$)>> - <<latex($\overline{X}$)>> )^2^ + ( <<latex($\overline{X}_{cocaine}$)>> - <<latex($\overline{X}$)>>)^2^]<<BR>>
+  <<latex($SS_{R}$)>> =  <<latex($n$)>> [( <<latex($\overline{X}_{control}$)>> - <<latex($\overline{X}$)>> )^2^ + ( <<latex($\overline{X}_{cocaine}$)>> - <<latex($\overline{X}$)>>)^2^]<<BR>>
-Line 213:
+Line 219:
-Both factors (substance and duration of usage) are significant, as indicate by the fact that '''F,,OBT,,''' > '''F,,CRIT,,'''. The interaction between the two (r * c) is also significant. To interpret the significant interaction we can look at our plot of group means to see this:
+Both factors (prenatal-exposure and age) are significant, as indicate by the fact that '''F,,OBT,,''' > '''F,,CRIT,,'''. Thus, we can reject all H,,01,, and H,,02,, and conclude that dendritic spine density is affected by prenatal exposure and age. The interaction between the two factors (r * c) is also significant. Thus, we can also reject H,,03,, and conclude there is a significant interaction between these two factors. 

 To further interpret these results, we can plot the group means as follows:
-Line 217:
+Line 225:
-So, as indicated above, in the placebo group (the control), the longer the squirrels are in the study, the more hours per night they sleep. (Maybe the study is exhausting, or depressing, but that's a question for a follow-up study... this is just an example problem with data we made up, so you needn't worry about that now!) But there is no such effect in the cocaine group. The cocaine-consuming squirrels sleep less in both groups, regardless of for how long they've been in the study. The interaction is that the study duration effect varies as a function of what substance group a squirrel is in.
+So, as indicated above, spine density is significantly greater in the control group as compared to drug-exposed animals at both 4 and 12 weeks of age. The effect of prenatal drug exposure is magnified with time due to a developmental increase in spine density that occurs only in the control group. Between 4 and 12 weeks of age, spine density in prefrontal cortex increases significantly in control animals, while it remains relatively unchanged in the drug-exposed animals.