INTRODUCTION
In poultry farming, the production of high quality (healthy and weighty) birds is always desired, as this boosts the revenue of the poultry farmer. To ensure this, the best of inputs (especially, poultry feeds) has to be used on the birds. This study is set out to show how profile analysis could be carried out and particularly, its application in the selection of poultry feed brands.
Profile analysis, according to Ott (1999), is a specific
style of Multivariate Analysis of Variance (MANOVA). It is equivalent to repeated
measures MANOVA (Tabacknick and Fidell, 2006), because
of its multiple responses taken in sequence on the same subject(s). Usually,
the responses are taken over time as in weekly weight measurements to establish
growth curves (Littell et al., 1998).
THEORY AND PROCEDURE OF PROFILE ANALYSIS
Let X_{ijk}, i = 1, 2, …, g; j = 1, 2, …, n_{i}; k = 1, 2, …, p, be an observation (a response) in a repeated measures experiment, where, i, j and k stand for treatment group (population), selected subject in the ith treatment group and observation (response) level, respectively.
Let,
denote the response vector for the jth subject within the ith treatment group
and denote the mean response vector for the ith treatment group.
Multivariate Analysis of Variance (MANOVA) is the natural choice for analyzing the type of data in the oneway completely randomized design described in Table 1, since it models the interdependencies among the response variables. However, profile analysis (Repeated Measures MANOVA) is most appropriate.
MANOVA with repeated measures is used when the measure that is repeated (e.g.,
across time) is a compound formed in the usual MANOVA fashion across multiple
dependent variables. Thus, it is different from having multiple repeated measures
factors (Tabacknick and Fidell, 2006).
It is worthy of note that certain assumptions and conditions Morrison
(2002) and Tabacknick and Fidell (2006) must be met
before profile analysis could be carried out on any set of data.
Table 1: 
Layout of Repeated Measures Design for gGroup Profile Analysis 

The question of equality of the population mean response vectors sought in
MANOVA which is given by:
H_{0} : μ_{1} = μ_{2} = … = μ_{g}

is divided into several specific possibilities. The question is formulated
in a stagewise fashion:
• 
Are the profiles parallel?
Or equivalently: H_{01}: μ_{1k}μ_{1(k1)}
= μ_{2k}μ_{2(k1)} = … = μ_{gk}
μ_{g(k1)}; k = 2, 3, …, p 
• 
Assuming the profiles are parallel, are they coincident?
Or equivalently: H_{02} :μ_{1k} = μ_{2k}
= … = μ_{gk}; k = 1, 2, …, p 
• 
Assuming the profiles are coincident, are they level? That is, are
all means equal to the same constant?
Or equivalently: H_{03}: μ_{11} = μ_{12}
= … = μ_{1p} = μ_{21} = μ_{22}
= … = μ_{2p} = … = μ_{g1} =μ_{g2}
= … = μ_{gp} 
Before testing the hypothesis of parallelism, it is advisable to make a plot
of the means (otherwise known as average profiles).
Test for parallel profiles: This test is achieved by working with the successive differences in the responses. The profiles are parallel if the differences are the same across the treatment groups. Thus, the null hypothesis in stage 1 can be written as:
or
H_{01}: Cμ_{1} = Cμ_{2}
= … = Cμ_{g} 
where, C is contrast matrix given by:
For independent samples of sizes, n_{1}, n_{2}, …, n_{g}
from the ggroups, H_{01} can be tested (in the usual oneway MANOVA
fashion) by constructing the contrasttransformed observation vectors:
CX_{ij}; i = 1, 2, …, g; j = 1, 2, …,
n_{i} 
having sample mean vectors:
C _{ij};
i = 1, 2,…,g 
and grand mean vector, C
There are several different multivariate test statistics available for the
test of parallel profile, all of which will generally yield equivalent results.
Amongst the four common test statisticsnamely Wilks Lambda, Pillai’s
Trace, HotellingLawley Trace and Roy’s Greatest Root; Wilks Lambda (Λ)
is the most desirable because it can be converted exactly to an Fstatistic
(Ott, 1999; Everitt and Dunn, 2001).
For details of this conversion and the exact distribution of Λ, see Johnson
and Wichern (2001). It also presented a modification of Λ due to Bartlett
(1938) for cases where the number of groups is more than three (g>3),
as well as when large sample sizes are involved. It is worthy of note here that
(p1) would replace p in the circumstances above.
Employing the Wilks lambda criterion, therefore, H_{01} is rejected
at α level if the ratio of generalized variances:
is too small, or rather still if its equivalent Fstatistic is greater than
the critical value. For instance, where (p1)≥1 and g = 3, the critical region
would be:
The matrices B_{1} and W_{1} in Eq. 4 are
the treatment (Between) sum of squares and cross products, respectively and
residual (Within) sum of squares and cross products for the CX_{ij}’s
given as:
and
If H_{01} is rejected, it would be concluded that at least one of the average profiles is significantly different. Consequently, it would rather be unreasonable to embark
on testing the second null hypothesis. Otherwise proceed with the test.
Test for coincident profiles: The second null hypothesis investigates whether the profiles are superimposed on one another or rather identical. Under the condition of parallel profiles will be coincident only if the total heights:
are equal (Johnson and Wichern, 2001). Therefore, the
null hypothesis at this stage can be written in the equivalent form:
H_{02}: 1^{T}μ_{1} =
1^{T}μ_{2} = … = 1^{T}μ_{g} 
Where:
Hence, the test is univariate, based on the univariate observations:
1^{T}X_{ij}, i = 1, 2, … , g; j
= 1, 2, … , n_{i} 
Invariably, this is equivalent to performing a oneway ANOVA on the subject
totals. Timm (1975) stated that the univariate and multivariate
tests are equivalent, assuming parallelism. Nevertheless, the multivariate test
approach is preferable since the data have already been arranged in a multivariate
configuration for the first test.
Employing the Wilks lambda criterion, H_{02} is rejected at α
level if the ratio of generalized variances:
is too small, or rather still if its equivalent Fstatistic is greater than
the critical value. Generally, the critical region for the null hypothesis of
coincident profiles is given by:
unless where a modification of Λ_{2} due to Bartlett is, however, sought.
The matrices B_{2} and W_{2} in Eq. 9 are,
respectively given as:
and
If H_{02} is rejected, it would be concluded that the profiles were not identical and embarking on testing the third null hypothesis would rather be unreasonable. Otherwise, proceed with the test.
Test for level profiles: The third null hypothesis investigates whether all variables have the same mean, so that the common profile is level. That is, μ_{1} = μ_{2} = … = μ_{p}.
A simple way to approach this problem is to again take successive differences,
as was done in the first test. If the profile is indeed level, then the (p1)
differences should be zero. Thus, the null hypothesis at this stage can be written
as:
where, C is as given by Eq. 3 and μ (the common mean
vector) is estimated by
(the sample grand mean vector).
Employing Wilks Lambda criterion, as before, H_{03} is rejected at
α level if the ratio of generalized variances:
is too small, or rather still if its equivalent Fstatistic is greater than
the critical value. Generally, the critical region for the null hypothesis of
level profiles is given by Ott (1999) as:
The matrix W_{3} in Eq. 13 is the same as W_{1}
in Eq. 7, while under H_{03} the between sum of squares
and crossproducts for the CX_{ij}’s is:
If H_{03} is rejected, the conclusion thus becomes that all the means are not equal to the same constant. Otherwise, the profiles are level.
Contrasts following profile analysis: The major description of result
in profile analysis, especially whenever the parallelism or levels hypothesis
is rejected, is typically the plot of average profiles. However, when there
are more than two levels of significant effects in profile analysis (in MANOVA,
generally), it is often desirable to perform contrasts (example, Dunnett’s,
Tukey, Bonferroni, or Scheffe’) to pinpoint sources of variability (Tabacknick
and Fidell, 2006).
MATERIALS AND METHODS
A practical example is presented on the application of profile analysis in analyzing the performances of three brands of poultry feed with a view to selecting the best brand.
Data collection and description: The data for this study were collected from the Planning, Research and Statistics Unit of Talented Royal Apex Project (TRAP), Owerri, Imo State of Nigeria. The data were the outcome of an experiment conducted to study the resultant weightyields of poultry birds upon consumption of three brands of poultry feed in Ndubuisi Farms (a subsidiary of TRAP), with a view to comparing a locallyproduced brand of poultry feed by TRAP (branded TRAP FEEDS) with two other brands (GUINEA FEEDS and TOP FEEDS).
Thirty dayold male turkey broilers were randomly selected from a batch of
fiftywith common featuresand put in three different poultry houses (ten in
each). The birds were properly identified (by numbers) and weighed (in grammes).
TOP, GUINEA and TRAP FEEDS were equitably administered to the birds in poultry
houses 1, 2 and 3, respectively, for the eightweek period. Also, conventional
feeding and health programmes obtainable in poultry farming were strictly observed.
Analysis: The data were analyzed by employing the profile analysis procedure
outlined earlier. First, as tradition demands, the data were checked and it
is sufficient to say that all the assumptions of profile analysis were satisfied.
Although there were less than 20 birds in each of the treatment groups (purposely
done to avoid overcrowding in the poultry houses), with equal sample sizes
(n_{1} = n_{2} = n_{3}) and n_{1}>p; i =
1, 2 and 3, the multivariate normality assumption is not likely to be violated.
Besides, profile analysis is as robust to violation of multivariate normality
assumption as other forms of MANOVA (Tabacknick and Fidell,
2006).
A plot of the average profiles for the weightyields of the birds in the three
treatment groups is constructed (Fig. 1) and then followed by the test for parallel
profiles.

Fig. 1: 
Average profiles of weekspecific weights poultry birde for three brands
of feeds 
Testing for parallel profiles: We have the hypothesis of parallelism
as:
H_{01} : Cμ_{1} = Cμ_{2}
= Cμ_{3} 
By using Eq. 6
and Eq. 7
Equation 4 yields Λ_{1} = 2.733E3 (using MATLAB
6.0), while with p = 8 and g = 3, Eq. 5 yields:
Consequently, H_{01} is rejected at the 5% level of significance.
RESULTS AND DISCUSSION
The result of the analysis showed that the profiles were not parallel and this caused the other two hypotheses (H_{02} and H_{03}) contingent on the tenability of H_{01}, not to be tested. In other words, significant differences existed amongst the profiles of the three treatment groups.
For profile analysis, the major description of results is typically a plot
of profiles. Figure 1 showed that the average profile for
TRAP FEEDS (Group 3) was well above the other two groups and this implied that
its average performance was the highest regarding the resultant weekly weightyields
of the birds upon consumption.
Furthermore, contrasts performed by the method of Scheffe’ showed that there was no significant difference between the profiles of TOP and GUINEA FEEDS while the profile of TRAP FEEDS was significantly different from those of TOP and GUINEA FEEDS.
Consequently, TRAP FEEDS was selected as the best brand of poultry feed amongst the three brands used in the farm.
CONCLUSION
In this study, we have presented the procedure of carrying out profile analysisthe
most appropriate statistical method for analyzing repeated measures data. We
have also demonstrated the application of profile analysis in analyzing the
performances of three brands of poultry feed, to know whether they are equal.
The analyses showed that the profiles were not parallel (that is, statistically
significant differences exist in the average performances of the feeds). The
average profile of TRAP FEEDS was found to be well above those of the other
two brands of feed, which implied highest performance. Hence, TRAP FEEDS was
selected as the best brand of poultry feed for the birds in the farm.