Background | Descriptive Statistics | Inferential Statistics | Sample Size Requirements | Exercises

In a previous chapter we considered the analysis of a continuous outcome from two independent groups. In this chapter we extendt his
method to consider 2 *or more *groups.

** Illustrative Data. **Let us consider ages of subjects from three centers. Data are:

Center 1: 60, 66, 65, 55, 62, 70, 51, 72, 58, 61, 71, 41, 70, 57, 55, 63, 64, 76, 74, 54, 58, 73

Center 2: 56, 65, 65, 63, 57, 47, 72, 56, 52, 75, 66, 62, 68, 75, 60, 73, 63, 64

Center 3: 67, 56, 65, 61, 63, 59, 42, 53, 63, 65, 60, 57, 62, 70, 73, 63, 55, 52, 58, 68, 70, 72, 45

The data set has 63 records (*N* = 63) and a grand mean of 62.1 years (standard deviation = 8.1 years).

For these data to be analyzed in *Epi Info*, they must be structured with two variables -- one for the dependent (outcome) variable and
one for the independent (group) variable. Data for the illustrative data set are stored in ** AGEBYCEN.ZIP** as the file

`REC AGE CENTER
--- --------- -----------
1 60 1
2 66 1
3 65 1
etc.
63 45 3 `

Summary statistics are computed with the `MEANS` command:

`EPI6> MEANS <DV> <IV>`

where `<DV>` represents the dependent variable and `<IV>` represents the independent variable.

To compute statistics for the illustrative data, issue the commands:

`EPI6> READ AGEBYCEN.REC`

`EPI6> MEANS AGE CENTER`

This produces the following output:

` MEANS of AGE for each category of CENTER`

`CENTER Obs Total Mean Variance Std Dev`

`1 22 1376 62.545 75.212 8.672`

`2 18 1139 63.278 60.683 7.790`

`3 23 1399 60.826 64.059 8.004`

`CENTER Minimum 25%ile Median 75%ile Maximum Mode`

`1 41.000 57.000 62.500 70.000 76.000 55.000`

`2 47.000 57.000 63.500 68.000 75.000 56.000`

`3 42.000 56.000 62.000 67.000 73.000 63.000`

Thus, *n*_{1} = 22, *n*_{2} = 18, and *n*_{3} = 23. We also see that groups have similar means (means are 62.5, 63.3, and 60.8, respectively) and
standard deviations are not too dissimilar.

Side-by-side quartile plots can be drawn (by hand) with minimal effort by graphing each group median as a dot and whiskers from the group's minimum to 25 percentile (bottom whisker) and 75 percentile to the maximum (top whisker). Click here for an example.

We assume the Mean Square Within (`MSW`) in the ANOVA table is an pooled estimate of the common within group variance:

` ANOVA`

`Variation SS df MS F statistic p-value`

`Between 66.614 2 33.307 0.497 0.616421`

`Within 4020.370 60 67.006`

The standard error of the mean for group *i* (*se _{i}*) is thus:

*se _{i}* =sqrt(

with df_{w} = *N*-*k*, where *N *represents the total sample size (all groups combined) and *k* represents the number of groups.

For the illustrative example, *se*_{1} = sqrt(67.006 / 22) = 1.745.

A 95% confidence interval for the mean of group __i____ (µ__* _{i}*) is given by:

(sample mean of group *i*) ± (*t _{df}*

For example, a 95% confidence for the mean of group 1 = 62.5 ± (*t*_{60,.025})(1.745) = 62.5 ± (2.00)(1.745) = (58.9, 66.1).

The objective of ANOVA is to determine whether one or more population means of the *k* groups differs. The null and alternative
hypotheses are:

*H*_{0}: µ_{1} = µ_{2} = ... = µ_{k}_{}

*H*_{1}: at least one population mean differs

where µ_{i}_{} represents the population mean of group *i* {*i*: 1, 2, . . . *k*}.

Briefly, ANOVA partitions the variance in the data into the variance or mea square between (`MSB`) and the variance or mean square
(`MSW`). The ratio of these means squares is the *F *statistic:

*F*_{stat} = (`MSB`) / (`MSW`)

Under the null hypothesis, this test statistic has an *F* distribution with *df*_{B} = *k*-1 and *df*_{W} = *N*-*k*. The test is one-tailed focusing on the
upper extent of the *F _{df}*

` ANOVA`

`Variation SS df MS F statistic p-value`

`Between 66.614 2`

Thus, *p* =.62.

** Assumptions: **The ANOVA tests has several hidden assumptions. Traditionally, we speak of the assumption of independence,
normality, and equal variance. In addition, statistical inferences assume validity of the data (i.e., freedom from selection bias and
information bias) and minimal confounding.

The Kruskal-Wallis procedure is a non-parametric analogue of ANOVA. The null and alternative are:

*H*_{0}: the population medians are
equal

H_{1}: at least one population median differs

Statistics are provided in the two-variable `MEANS`
command output:

` Kruskal-Wallis One Way Analysis of Variance`

`Kruskal-Wallis H (equivalent to Chi square) = 0.916`

` Degrees of freedom = 2`

` p value = 0.632634`

The results of this test (*p* = .63) provide no reason to reject *H*_{0}.

Bartlett's test address whether population variances differ. The null and alternative hypotheses are:

*H*_{0}: s²_{1} = s²_{2} = . . . = s²_{k}_{}

*H*_{1}: at least one population variance differs

where s²_{i}_{} represents the population variance in group *i*. Results are provided
in the output of the two-variable `MEANS` command. Output for
the illustrative example is:

` Bartlett's test for homogeneity of variance`

`Bartlett's chi square = 0.243 deg freedom = 2 p-value = 0.885608`

Thus, for this example, c^{2} = 0.24 with
2 df (*p* = .89), providing no real evidence against the null
hypothesis.

Comment: Because Bartlett's test performs poorly in non-normal populations and has poor
power some statisticians advise against its routine use
(Box, 1953, *Biometrika*, pp. 318 - 335).

We frequently want to know how large a sample is needed when testing *k*
means. Although there is no simple answer to this question, a reasonable sample
size can be determined if certain assumptions are made. Let us concern ourselves with trying to establish a significant difference
among *k *means (via ANOVA) by asking how big a sample size is needed
to (a) detect a difference between two means of D,
(b) at a type I error
rate of a, (c) with probability (power) 1-b.
It is necessary to have a prior estimate of variability s of the
outcome variable, with such estimates coming from a pilot study, prior published
results, a preliminary
analysis, or intuition. Computational solutions are possible once these
underlying assumptions are made clear, with formulas are available in Sokal & Rohlf,
1996, pp. 263-264 (for instance). Calculations have been programmed into a the
Dept of OB/GYN at the University of Hong Kong website via the URL http://department.obg.cuhk.edu.hk/ResearchSupport/Sample_size_CompMean.asp. (If
this link does not take you directly to the sample size calculator, click `Sample
Size > Comparing Means`.)

** Illustrative example. **Suppose we test

Type I error=0.05 | Type I error=0.01 | Type I error=0.001 | |
---|---|---|---|

Power=80% | 41 | 60 | 87 |

Power=90% | 54 | 76 | 107 |

Power=95% | 67 | 91 | 125 |

Notice that the output provides samples sizes per group (*n _{i}*)
at various power and alpha levels. For example, under the stated assumptions, we need

**(1) ALCOHOL.ZIP**: *Alcohol Consumption by Income Level *(Data
from Monder, 1986). Data come from a survey of alcohol
consumption and socioeconomic status. Data, in ALCOHOL.REC, are coded as follows:

Variable Name | Type | Description and codes |

ALCS | ## | Alcohol consumption score. Codes are as follows: 00 = non-drinker 01 = 1 drink per week 02 = 1-2 drinks per week 03 = 2 drinks per week 04 = 2-3 drinks per week 05 = 3 drinks per week 06 = 3-4 drinks per week 07 = 4 drinks per week 08 = 4-5 drinks per week 09 = 5 drinks per week 10 = 5-6 drinks per week 11 = 6 drinks per week 12 = 7-11 drinks per week 13 = 12+ drinks per week |

AGE | ## | Age (in years) |

INC | # | Income level: 1 = low, 5 = high |

**(A) Univariate description of ALCS. **Before performing ANOVA, describe the distribution of alcohol scores for all groups combined
(`MEANS ALCS`). Show the distribution of this variable in the form of a histogram (`HISTOGRAM ALCS)`. Are data skewed? What
percentage of people in the sample are non-drinkers?

**(B) ALCS by INC. **Assess alcohol consumption by income.

**(2) DEERMICE**: *Weight Gain in White-Footed Deer Mice* (Hampton, 1994, p. 118, modified). Fifteen deermice are randomly
assigned to one of three groups. Group A receives a standard diet, Group B receives a diet of junk food, and Group C receives a diet of
health food. The research question is to determine whether `WTGAIN `differs by `DIET`. Weight gains (gms.) are as follows:

Group A: 11.8, 12.0, 10.7, 9.1, 12.1

Group B: 13.6, 14.4, 12.8, 13.0, 13.4

Group C: 9.2, 9.6, 8.6, 8.5, 9.8

**(3) ROOSTER: ***Testosterone Levels in Roosters *(Data from Hampton, 1994, p. 147). A chicken pathologist believes testosterone
levels differ by rooster strain. To test this hypothesis, testosterone levels are measured in 3 strains of roosters. The research question is
to determine whether testosterone levels differ by rooster strain. Data are:

REC TESTOSTERO STRAIN

--- ---------- ------

1 439 A

2 568 A

3 134 A

4 897 A

5 229 A

6 329 A

7 103 B

8 115 B

9 98 B

10 126 B

11 115 B

12 120 B

13 107 C

14 99 C

15 102 C

16 105 C

17 89 C

18 110 C

**(4) MAT-ROLE.ZIP: ***Adaptation to Maternal Roles *(Howell, 1995, pp. 302 - 304). In a study of the development of low-birthweight
(LBW) infants, three groups of newborns differed in terms of birthweight and whether their mothers had participated in a training
program about the special needs of low-birthweight infants. The mothers were then interviewed with the infants were 6 months old.
There were three groups in the experiment: an LBW Experimental group (Group 1), an LBW Control group (Group 2), and a
Full-Term Control group (Group 3). The two control groups received no special training, and so serve as a reference against which to
compare the performance of the experimental intervention. The LBW Experimental group was part of the intervention program, and
the researchers hoped to show that those mothers would adapt to their new role as well as the mothers of full-term infants. On the other
hand, they expected mothers of LBW infants who did not receive the intervention to have some difficulty adapting. The outcome
measure is an adaptation scale, whereby high values indicate some trouble adapting. (Being a parent of a low-birthweight baby is not
an easy task, especially for the first few months, see Achenbach et al. 1993). Data are contained in MAT-ROLE, which can be
downloaded form the server by clicking on the highlight filename, above. Download the data set and analyze these data.