3 Two-way ANOVA versus one-way ANOVA

You should note that a two-way ANOVA design is rather more efficient than a one-way design. In the last example, we could fix the testing station and look at the electronic assemblies produced by a variety of machines. We would have to replicate such an experiment for every testing station. It would be very difficult (impossible!) to exactly duplicate the same conditions for all of the experiments. This implies that the consequent experimental error could be very large. Remember also that in a one-way design we cannot check for interaction between the factors involved in the experiment. The three main advantages of a two-way ANOVA may be stated as follows:

  1. It is possible to simultaneously test the effects of two factors. This saves both time and money.
  2. It is possible to determine the level of interaction present between the factors involved.
  3. The effect of one factor can be investigated over a variety of levels of another and so any conclusions reached may be applicable over a range of situations rather than a single situation.
Exercises
  1. The temperatures, in Celsius, at three locations in the engine of a vehicle are measured after each of five test runs. The data are as follows. Making the usual assumptions for a two-way analysis of variance without replication, test the hypothesis that there is no systematic difference in temperatures between the three locations. Use the 5% level of significance.
    Location Run 1 Run 2 Run 3 Run 4 Run 5
    A 72.8 77.3 82.9 69.4 74.6
    B 71.5 72.4 80.7 67.0 74.0
    C 70.8 74.0 79.1 69.0 75.4
  2. Waste cooling water from a large engineering works is filtered before being released into the environment. Three separate discharge pipes are used, each with its own filter. Five samples of water are taken on each of four days from each of the three discharge pipes and the concentrations of a pollutant, in parts per million, are measured. The data are given below. Analyse the data to test for differences between the discharge pipes. Allow for effects due to pipes and days and for an interaction effect. Treat the pipe effects as fixed and the day effects as random. Use the 5% level of significance.
    Day
    Pipe A
    1 160 181 163 173 178
    2 175 170 219 166 171
    3 169 186 179 178 183
    4 230 206 216 195 250
    Day
    Pipe B
    1 172 164 186 185 172
    2 177 170 156 140 155
    3 193 194 189 156 181
    4 212 235 195 206 209
    Day
    Pipe C
    1 214 196 207 219 200
    2 186 184 181 189 179
    3 209 220 199 185 228
    4 254 293 283 262 259
  1. We calculate totals as follows.

    Run Total Location Total
    1 215.1 A 377.0
    2 223.7 B 365.6
    3 242.7 C 368.3
    4 205.4 Total 1110.9
    5 224.0
    Total 1110.9

    ∑ ∑ y i j 2 = 82552.17

    The total sum of squares is

    8255217 − 1110 . 9 2 15 = 278.916 on 15 − 1 = 14 degrees of freedom.

    The between-runs sum of squares is

    1 3 ( 215 . 1 2 + 223 . 7 2 + 242 . 7 2 + 205 . 4 2 + 224 . 0 2 ) − 1110 . 9 2 15 = 252.796

    on 5 − 1 = 4 degrees of freedom.

    The between-locations sum of squares is

    1 5 ( 377 . 0 2 + 365 . 6 2 + 368 . 3 2 ) − 1110 . 9 2 15 = 14.196 on 3 − 1 = 2 degrees of freedom.

    By subtraction, the residual sum of squares is

    278.916 − 252.796 − 14.196 = 11.924 on 14 − 4 − 2 = 8 degrees of freedom.

    The analysis of variance table is as follows.

    Source of variation Sum of squares Degrees of freedom Mean square Variance ratio
    Runs 252.796 4 63.199
    Locations 14.196 2 7.098 4.762
    Residual 11.924 8 1.491
    Total 278.916 14

    The upper 5% point of the F 2 , 8 distribution is 4.46. The observed variance ratio is greater than this so we conclude that the result is significant at the 5% level and reject the null hypothesis at this level. The evidence suggests that there are systematic differences between the temperatures at the three locations. Note that the Runs mean square is large compared to the Residual mean square showing that it was useful to allow for differences between runs.

  2. We calculate totals as follows.
    Day 1 Day 2 Day 3 Day 4 Total
    Pipe A 855 901 895 1097 3748
    Pipe B 879 798 913 1057 3647
    Pipe C 1036 919 1041 1351 4347
    Total 2770 2618 2849 3505 11742

    ∑ ∑ ∑ y i j k 2 = 2356870

    The total number of observations is N = 60.

    The total sum of squares is

    2356870 − 1174 2 2 60 = 58960.6

    on 60 − 1 = 59 degrees of freedom.

    The between-cells sum of squares is

    1 5 ( 85 5 2 + ⋯ + 135 1 2 ) − 1174 2 2 60 = 58960.6

    on 12 − 1 = 11 degrees of freedom, where by “cell” we mean the combination of a pipe and a day.

    By subtraction, the residual sum of squares is

    58960.6 − 48943.0 = 10017.6

    on 59 − 11 = 48 degrees of freedom.

    The between-days sum of squares is

    1 15 ( 277 0 2 + 261 8 2 + 284 9 2 + 350 5 2 ) − 1174 2 2 60 = 30667.3

    on 4 − 1 = 3 degrees of freedom.

    The between-pipes sum of squares is

    1 20 ( 374 8 2 + 364 7 2 + 434 7 2 ) − 1174 2 2 60 = 14316.7

    on 3 − 1 = 2 degrees of freedom.

    By subtraction the interaction sum of squares is

    48943.0 − 30667.3 − 14316.7 = 3959.0

    on 11 − 3 − 2 = 6 degrees of freedom.

    The analysis of variance table is as follows.

    Source of variation Sum of squares Degrees of freedom Mean square Variance ratio
    Pipes 14316.7 2 7158.4 10.85
    Days 30667.3 3 10222.4 48.98
    Interaction 3959.0 6 659.8 3.16
    Cells 48943.0 11 4449.4 21.32
    Residual 10017.6 48 208.7
    Total 58960.6 59

    Notice that, because Days are treated as a random effect, we divide the Pipes mean square by the Interaction mean square rather than by the Residual mean square.

    The upper 5% point of the F 6 , 48 distribution is approximately 2.3. Thus the Interaction variance ratio is significant at the 5% level and we reject the null hypothesis of no interaction. We must therefore conclude that there are differences between the means for pipes and for days and that the difference between one pipe and another varies from day to day. Looking at the mean squares, however, we see that both the Pipes and Days mean squares are much bigger than the Interaction mean square. Therefore it seems that the interaction effect is relatively small compared to the differences between days and between pipes.