two way anova

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					                                 TWO WAY ANOVA

We have dealt with the simplest form of ANOVA a 1-way ANOVA which means there is
only one variable. It should come as no surprise then that using ANOVA for two
variables is called a 2-way ANOVA! Again each variable can have a number of different
conditions or levels.

Here is some example data from Hinton (1995). The study involved looking at an
expanding company who wanted to introduce a new machine into their factory. They
were concerned with whether they should transfer staff working on the old machine to the
new machine or whether they should employ new staff to operate it.

A psychologist selected 12 staff who had experience of the old machine, and 12 staff who
had no such experience. Half the participants from each group were allocated to the old
machine and half to the new. The number of errors made by the subjects over a set time
period was measured.

Here is the data set:

Experience       Old         New
Novice           4           5
                 5           6
                 7           5
                 6           6
                 8           5
                 5           6
Experienced      1           8
                 2           9
                 2           8
                 3           8
                 2           7
                 3           9

As usual with any data set we should try to look at it in its raw form (if there are too
many items of data this is difficult, but we can manage it here with only 24 data points).
It is noticeable straight away that the lowest values in the data set appear in the
experienced staff working on the old machine. We should have expected this, these
people are very familiar with the old machine and therefore make fewer mistakes. If we
compare this cell with that of the novice staff working on the old machine we see that the
novices make more mistakes. Again, this may have been expected as they were new to
the old machine.

In comparing the novice staff to the old staff this time in terms of performance on the
new machine we find a different and perhaps less predictable story. The old staff make a
few more mistakes on the new machine than do the new staff. We might hypothesise that
 this is because they have carried over learned habits from operating the old machine
 which are not appropriate for the new machine.
 Here is another way of viewing these results using another 2 x 2 table with the means.

 Machine        Old            New
 Novice         5.83           5.50
 Experienced    2.16           8.16

 Here we can see the pattern expressed in terms of the means from the four conditions.
 This of course also shows that the lowest number of mistakes were, made by experienced
 people on the old machine, and the highest were also by this group on the new machine.

Machine        Old     New        Total
Novice         5.83    5.50       11.33
Experienced    2.16    8.16       10.34
Total          8.00    13.66

 We can see that summarising for the variable “machine” (see underlined values in figure
 above) M=8 for old and M=13.66 for new machine.
 Summarising the other variable “experience” there is less of a difference (see underlined
 values in figure below) M=11.33 for novice and M=10.34.
Machine Old             New         Total
Novice       5.83       5.50        11.33
Experien 2.16           8.16        10.34
Total        8.00       13.66
 So the effect is more pronounced for the machine than for the type of participant. When
 referring to the experimental effect (the difference between scores in conditions or levels
 of the same variable) we call this a “main effect”. It is possible here that we have a main
 effect for the type of machine but much less so for the experience of the staff.
 When you have more than one variable you can have interactions between the variables
 (where the levels of a variable relates systematically levels of another variable).
 Interactions are easier to see if you present the results in a graphical form. Below is a
 mean interaction plot for the machine data.
8                                            New
                  Nov              Exp

The lines crossing over each other clearly indicate an interaction. Here it is apparent that
performance in terms of number of mistakes increases for experienced staff as they move
from the old machine to the new, while it stays relatively constant for the new staff.

Finally let‟s carry out the calculation using SPSS and get the full ANOVA summary

                              Tes ts of Be tw ee n-Subje cts Effects

    Dependent Variable: errors
                             Ty pe III Sum
    Sourc e                  of Squares        df        Mean Square     F       Sig.
    Correc ted Model             109.833a            3        36.611    40.679     .000
    Intercept                    704.167             1       704.167   782.407     .000
    mac hine                      48.167             1        48.167    53.519     .000
    ex perience                      1.500           1         1.500     1.667     .211
    mac hine * experienc e        60.167             1        60.167    66.852     .000
    Error                         18.000            20          .900
    Total                        832.000            24
    Correc ted Total             127.833            23
      a. R Squared = .859 (Adjus ted R Squared = .838)

Here we have not one F value to inspect but three. Here two of these are significant; one
for the variable/factor „type of machine‟ and one for the interaction between
variables/factors „experience of staff‟ and „type of machine‟.

We can report then that:

There is no significant main effect for experience on the old machine (F(1,20)=1.67,
p=0.21), but there is a main effect for the type of machine (F(1,20)=53.52, p<0.01). In
addition, there is an interaction between these two variables (F(1,20)=66.85, p<0.01).

Where you obtain an interaction this should take precedence as the most interesting
finding. Your discussion should address possible reasons for this effect Here we might
assume that the reason for this is that the experienced members of staff bring old habits
(ways of using the old machine) to the use of the new machine and therefore make
mistakes as a consequence.

2 x 3 ANOVA

Here is another example of a multivariate ANOVA, this time a 2x3 rather than a 2x2.
You can still call it a 2-way ANOVA because it has variables but this time instead of
each of the two variables possessing 2 levels, one has 2 and one has 3, hence the 2x3. It
sounds more complicated than it really is!

This example is based on a fictitious data set used by Lindman (1974). It concerns the
nature-nurture question in the context of rats running a T-maze. A T-maze is a simple
maze and the rats are expected to learn to run straight to the food, placed in a particular
location, without errors. Three strains of rats were used which were considered to be
bright, dull or mixed in mental ability. From each of these strains, four animals were
reared in a free (stimulating) environment and four in a restricted environment. The data
recorded was the number of errors made while the rat ran the maze.

A few questions first:

? What are the independent variables?
1.            and levels
2.            and levels

? What is the dependent variable?

Here is the data set:

Env        Bright        Mixed   Dull
Free       26            41      36
           14            82      87
           41            26      39
           16            86      99
Restrict   51            39      42
ed         35            114     133
           96            104     92
           36            92      124

? Calculate the means for this data presenting them in a 2x3 table.
? What do the means suggest?

? Draw a mean interaction plot (on the other side of this paper) and interpret it.

Here is the full summary table for the ANOVA.

Eff      D   Mseff Df      Mserr F          p
Env      1   5551    18    953     5.8      .02
Strain   2   3969    18    953     4.1      .03
Env*     2   8.0     18    953     .008     .99

? Highlight the main values of interest and report the findings in a sentence or two,
explaining the results in relation to the variables