Two-way analysis of variance

In section 5.2 we introduced one-way ANOVA for testing hypothesis involving three or more population means. This ANOVA method is appropriate as long as we are interested in analyzing one factor at a time and we select independent random samples from the populations. There are situations in which another factor affects the observed response in a one-way design.

Suppose that we have K groups and H blocks. We will use to denote the sample observations corresponding to the group and the block, as it shown in Figure 5.2. There are total

number of observations.

Our aim is to test the null hypothesis that all group means are equal, and the null hypothesis that all block means are equal.

To develop these tests we need to set two-way ANOVA table.

BLOCK GROUP

1 2 …… K

1 ……. 2 ……. . … … ……. ….. . …. …. ….. … H …….

Figure 5.2

To set this table we will use following steps:

1) Find sample mean for each group. For the mean of the group we use notation , defined as

;

2) Find sample mean for each block. The mean of the block we use notation , defined as

;

3) Find the overall mean of the sample observations. The overall mean denoted , defined as

4) Find between groups sum of squares, denoted , defined as

5) Find between blocks sum of squares, denoted , defined as

6) Find the error sum of squares, denoted , defined as

7) Find the total sum of squares, denoted , defined as

It can be shown that

8) We define the following mean squares

Between-groups:

Between-blocks:

Error:

9) We define two F ratios

and

We will use ratios above to test the null hypothesis about equality of population blocks and population groups.

1) The null hypothesis that the K population group means are the same is provided by the decision rule

Reject if

2) The null hypothesis that the H population block means are the same is provided by the decision rule

Reject if

where, is the number exceeded with probability by a random variable following an F distribution with numerator degrees of freedom and denominator degrees of freedom .

It is very convenient to summarize the calculations in tabular form, called a two-way analysis of variance table or ANOVA table, shown below (Table5.2):

Table 5.2

Source of Sum of Degrees of Mean variation squares freedom squares	F ratios
Between groups SSG Within blocks SSB Error SSE Total SST

Exercise:

Four drivers tested three types of cars for fuel consumptions. The accompanying table shows fuel consumptions of cars

Block (Drivers) Group (Cars)

A B C

1 22 24 26 2 21 25 22 3 19 20 23 4 18 19 21

a) Set out the two-way analysis of variance table.

b) Test the null hypothesis that the population mean fuel consumption is the same for all three types of cars. Take .

c) Test the null hypothesis that population values of mean fuel consumption are the same for each driver. Take .

Solution:

a)

1) Let us find sample mean for each group

2) Find sample mean for each block.

;

;

;

3) Find the overall mean of the sample observations.

4) Find between groups sum of squares

5) Find between blocks sum of squares

6) Let us find total sum of squares

68.67

7) By subtraction we obtain error sum of squares

8) For the fuel consumption data, the mean squares are

9) We define two F ratios

and

Source of Sum of Degrees of Mean variation squares freedom squares	F ratios
Between groups 18.67 2 9.34 Within blocks 38.71312.90 Error 11.296 1.88 Total 68.67 11	4.97 6.86

b) We can write the null hypothesis that the population means fuel consumption is the same for all three types of cars as

The decision rule is

Reject if

Since the value of test statistic 4.97 is not greater than 5.14, we fail to reject the null hypothesis. Therefore, we accept the hypothesis that the fuel consumptions are the same for all types of cars.

c) We write the null hypothesis of equality of the population values of mean fuel consumption for all four drivers as

The decision rule is

Reject if

Since the value of test statistic 6.86 is greater than 4.76, we reject the null hypothesis. Therefore, we accept the hypothesis that the fuel consumptions are not the same for each driver age class. In other words, fuel consumption of car depends on driver’s habit.

Remark1:

To use MINITAB menu follow the following instructions:

1. Select Stat>ANOVA>Two-way

2. Enter Response variable

3. Enter row factor

4. Enter column factor

5. Click OK.

Remark2:

For example above the MINITAB instruction is shown below

C1	C2	C3
Driver	Car	Fuel consumption

In this case the row factor is “Car” and the column factor is “driver”

Дата добавления: 2015-08-05