#
Independent Group ANOVA

(**One Way Analysis of
Variance**)

**Definition:** An extension of the independent group t-test where you have more
than two groups. Used to compare the means of more than two independent groups. This is
also called a One Way Analysis of Variance.

**Assumptions:** Subjects are randomly assigned to one of n groups. The distribution
of the means by group are normal with equal variances. Sample sizes between groups do not
have to be equal, but large differences in sample sizes by group may effect the outcome of
the multiple comparisons tests.

Test: The hypotheses for the comparison of independent groups are: (k is the number of
groups)

H_{o}: u_{1} = u_{2} _{... }= u_{k} (means of
the all groups are equal)

H_{a}: u_{i} <> u_{j} (means of the two or more groups are not
equal)

The test is performed in an Analysis of Variance (ANOVA) table. The test statistic is
an F test with k-1 and N-k degrees of freedom, where N is the total number of subjects. A
low p-value for this test indicates evidence to reject the null hypothesis in favor of the
alternative. In other words, there is evidence that at least one pair of means are not
equal.

**Multiple Comparisons:** Since the rejection of the null hypothesis does not
specifically tell you which means are difference, a multiple comparison test is often
performed following a significant finding in the One-Way ANOVA. There are a number of
multiple comparison procedures in the literature. Three that are available in WINKS and
KWIKSTAT are Newman-Keuls, Tukey, and Scheffé. A specialized multiple comparison,
Dunnett's test, is also available. Dunnett's test is used when the comparisons are
performed only with the control group versus all other groups.

Multiple comparison test are performed at a fixed significance level. In WINKS and
KWIKSTAT, that level is set a 0.05.

**Graphical comparison: **The graphical comparison allows you to visually see the
distribution of the groups. If the p-value is low, chances are there will be little
overlap between the two or more groups. If the p-value is not low, there will be a fair
amount of overlap between all of the groups. There are a number of options available in
the comparison graph to allow you to examine the groups. These include box plots, means,
medians, and error bars.

**Location in KWIKSTAT and WINKS: **The independent group t-test is located in the
Analyze/t-test and ANOVA menu.

**See Also:** When data are not normally distributed, The Kruskal-Wallis test, a
non-parametric test between groups, can be used. This test is available as an option in
the Nonparametric Comparisons menu.

**
****Example: Independent Group ANOVA (One-Way Analysis of variance**

The HOGDATA.DBF file on disk contains information on four different feeds and weight of
hogs after they had been fed one of the feeds for a period of time. The data are entered
in two fields, a GROUP field and OBS (Weight) field. The ANOVA results are:

**-----------------------------------------------------------------------------
Independent Group Analysis C:\WINKS\HOGDATA.DBF
-----------------------------------------------------------------------------**

**Grouping variable is GROUP
Analysis variable is OBS**

**------------------------------------------------------------------------------
Group Means and Standard Deviations
------------------------------------------------------------------------------**

**1: mean = 61.025 s.d. = 5.0823 n = 4
2: mean = 78.125 s.d. = 1.4886 n = 4
3: mean = 89.0667 s.d. = 4.4274 n = 3
4: mean = 85.775 s.d. = 3.5975 n = 4**

**Analysis of Variance Table**

**Source S.S. DF MS F Appx P
-----------------------------------------------------------------------------
Total 1923.41 14
Treatment 1761.24 3 587.08 39.82 <.001
Error 162.17 11 14.74**

**Error term used for comparisons = 14.74 with 11 d.f.**

** Critical q
Newman- Keuls Multiple Comp. Difference P Q (.05)
------------------------------------------------------------------------------
Mean(3)- Mean(1) = 28.0417 4 13.523 4.256 *
Mean(3)- Mean(2) = 10.9417 3 5.277 3.82 *
Mean(3)- Mean(4) = 3.2917 2 1.587 3.113
Mean(4)- Mean(1) = 24.75 3 12.892 3.82 *
Mean(4)- Mean(2) = 7.65 2 3.985 3.113 *
Mean(2)- Mean(1) = 1 7.1 2 8.907 3.113 ***

**Homogeneous Populations, groups ranked**

**Gp Gp Gp Gp
1 2 4 3
-----
--
--**

This is a graphical representation of the Newman- Keuls multiple comparisons.
Groups underscored by the same line are not significantly different.

In this output, the test statistic, F, is reported in the analysis of variance table.
The p-value for the F(3,11) = 39.82 is < 0.001. This means that there is evidence that
there are differences in the means across groups.

To determine what
specific means are different, read the results of the multiple comparison table. In this
case (using the Newman-Keuls procedure), differences were found between groups 3 and 1, 3
and 2, 4 and 1, 4 and 2, and 2 and 1. There was no significant difference found between
groups 4 and 3. Thus, feeds 3 and 4 both produce better results than feeds 1 and 2, but
there are not significantly different from one another.

The comparison graph allows you to visualize the difference between the groups.

**
****Exercise - Independent Group ANOVA (One Way Analysis of variance)**

Three different milling machines were being considered for purchase by a manufacturer.
Potentially, the company would be purchasing hundreds of these machines, so it wanted to
make sure it made the best decision. Initially, five of each machine were borrowed, and
each was randomly assigned to one of 15 technicians (all technicians were similar in
skill). Each machine was put through a series of tasks and rated using a standardized
test. The higher the score on the test, the better the performance of the machine. The
data are:

Machine 1 |
Machine 2 |
Machine 3 |

24.5 |
28.4 |
26.1 |

23.5 |
34.2 |
28.3 |

26.4 |
29.5 |
24.3 |

27.1 |
32.2 |
26.2 |

29.9 |
30.1 |
27.8 |

1. Perform an ANOVA on this data. Remember that the program expect two fields, a Group
field and a Observation field. Are there differences between the means by group?

2. What specific machines, if any, are different? Which machine would you recommend for
purchase?

3. Produce a graphical comparison of these results.