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The analysis of variance (ANOVA) is a generic statistical technique that assesses how the mean value of a variable is a↵ected by the classification of the data according to di↵erent sources of variation. The one-way ANOVA is a generalisation of the two-sample t-test, appropriate for any number of groups, and is equivalent to the two-sample t-test when assessing two groups.

If thep=0.05 (5%) level of significance is consistently accepted, then a wrong conclusion will be drawn on average once in every 20 tests performed, known as Type I error (rejecting theH0when it should be accepted, i.e. a false positive). If we then

perform multiple t-tests across a range of di↵erent treatments, than the probability of drawing at least one false conclusion is greatly increased. This probability could be reduced by lowering the significance level top=0.01 (1%), however, this increases the risk of making a Type II error (failing to reject the H0, i.e. a false negative).

Analysis of variance (ANOVA) overcomes these problems by comparing any number of sample means within a single test.

The null hypothesis for ANOVA is: H0 = the samples are drawn from nor-

mally distributed populations with equal means and variances, compared to the al-

ternative hypothesis: H1 = the samples are drawn from populations with di↵erent means.

The test statistic can be calculated as follows:

F = T reatM S ResM S , (2.10) where: T reatM S= n Pm j=1(¯xj x¯)2 m 1 , (2.11) ResM S= Pm j=1 Pn i=1(xji xj¯ )2 m(n 1) , (2.12)

Source of variation

Degrees of freedom

Sum of squares Mean square Variance ratio

Treatment (m-1) nP(¯xj x¯)2 TreatSS / (m-1) TreatMS /

ResMS Residual m(n-1) P P(xji x¯j)2 ResSS / (m(n-1))

Total n m - 1 P P(xji x¯)2

Table 2.6: One-way analysis of variance table, where TreatSS refers to the treatment sum of squares, TreatMS refers to the treatment mean square, ResSS refers to the residual sum of squares and ResMS refers to the residual mean square. To complete the test, the variance ratio was compared with tabulated critical values of the F- distribution to test the null hypothesis for the ANOVA.

The treatment mean square (TreatMS) is calculated by dividing the treat- ment sum of squares (the sum of squared deviations of the treatment (sample) means multiplied by the number of observation per sample) by its degrees of freedom (m- 1). The residual mean square (ResMS) is calculated by dividing the residual sum of squares (the sum of squared deviations of the individual observations about their respective sample means) by its degrees of freedom (m(n-1)).

The treatment sum of squares and residual sum of squares add up to the total sum of squares, which is the sum of the squared deviations of the individual obser- vations about the overall mean. These values allow the construction of an ANOVA table, which summarises all of the di↵erent components of variance, assuming equal replication, such as that shown in Table 2.6. To complete the test, the variance

ratio was compared with tabulated critical values of theF-distribution to test the

null hypothesis for the ANOVA. 2.5.6.1 Assumptions in ANOVA

• The residuals (observations minus the treatment mean) should be approxi- mately normally distributed;

• The residuals should have a common variance, which is not related to the treatment or magnitude of the mean response;

• The observations are independent.

A two-way ANOVA is a generalisation of the paired t-test, which allows the assessment of more than two treatments. This test looks at the grouping of exper- imental units into blocks, with the same number of experimental units per block. We can use ANOVA to calculate the treatment sum of squares, the block sum of

Source of variation

Degrees of freedom

Sum of squares Mean square Variance ratio

Blocks (b-1) tP(¯xi x¯)2 BlkSS / (b-1) BlkMS /

ResMS Treatment (t-1) bP(¯xj x¯)2 TreatSS / (t-1) TreatMS /

ResMS Residual (b-1)(t-1) by subtraction ResSS / ((b-1)(t-1))

Total b t - 1 P P(xji x¯)2

Table 2.7: Two-way analysis of variance table without interactions, where BlkSS refers to the block sum of squares, BlkMS refers to the block mean square, TreatSS refers to the treatment sum of squares, TreatMS refers to the treatment mean square, ResSS refers to the residual sum of squares and ResMS refers to the residual mean square. To complete the test, the variance ratio was compared with tabulated critical values of the F-distribution to test the null hypothesis H0 = there is no

di↵erence between blocks or treatments.

squares (the sum of squared deviations of the block means about the overall mean) and the total sum of squares. The residual sum of squares is usually calculated by subtraction as follows:

Residual SS = Total SS - (Block SS + Treatment SS)

The mean squares (variance) can be calculated by dividing the sum of squares by their respective degrees of freedom, and are used to test for di↵erences between treatments of blocks in the same way as the one-way ANOVA, such as that shown in Table 2.7. To complete the test, thevariance ratio was compared with tabulated critical values of the F-distribution to test the null hypothesis H0 = there is no

di↵erence between blocks or treatments.

Theanovan function was implemented inmatlab for both the diary and core

datasets, to assess the main e↵ects of altitude and individual on each of the mea- sured biochemical metabolites and physiological measurements. As there are no repeated measures available for the core dataset, we cannot assess the interactions between individual and altitude in a formal way, therefore a two-way ANOVA with- out interactions was performed. If the null hypothesis was rejected, then t-tests were performed between adjacent altitudes, to assess exactly which altitudes were significantly di↵erent from one another.