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Find a significant difference between groups Multivariate Analysis of Variance (MANOVA) Consider Univariate ANOVA Used when you have 3 or more samples C B A 508 514.25 583.25 727.5

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SLIDE 1

Multivariate Analysis of Variance (MANOVA)

Find a significant difference between groups

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SLIDE 2

C B A 𝑦 𝐡𝐢𝐷

𝐼𝑝: 𝜈𝐡 = 𝜈𝐢 = 𝜈𝐷 𝐼𝑏: 𝜈𝐡 β‰  𝜈𝐢 β‰  𝜈𝐷

The alternative could be true because all the means are different or just one

  • f them is different than the others

If we reject the null hypothesis we need to perform some further analysis to draw conclusions about which population means differ from the others and by how much 𝑦 𝑑 𝑦 𝐢 𝑦 𝐡

508 514.25 727.5 583.25

Consider Univariate ANOVA

Used when you have 3 or more samples

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SLIDE 3

C B A 𝑦 𝐡𝐢𝐷

Used when you have 3 or more samples

𝐺 = π‘‘π‘—π‘•π‘œπ‘π‘š π‘œπ‘π‘—π‘‘π‘“ 𝐺 = π‘€π‘π‘ π‘—π‘π‘œπ‘‘π‘“ 𝑐𝑓𝑒π‘₯π‘“π‘“π‘œ π‘€π‘π‘ π‘—π‘π‘œπ‘‘π‘“ π‘₯π‘—π‘’β„Žπ‘—π‘œ π‘€π‘π‘ π‘—π‘π‘œπ‘‘π‘“ 𝑐𝑓𝑒π‘₯π‘“π‘“π‘œ = 𝑦 𝑗 βˆ’ 𝑦 𝐡𝑀𝑀 2

π‘œ 𝑗

π‘œ βˆ’ 1 βˆ— 𝑠 π‘€π‘π‘ π‘—π‘π‘œπ‘‘π‘“ π‘₯π‘—π‘’β„Žπ‘—π‘œ = π‘€π‘π‘ π‘—π‘π‘œπ‘‘π‘“π‘—

π‘œ 𝑗

π‘œ

SIGNAL NOISE

A large F-value indicates a significant difference 𝑦 𝑑 𝑦 𝐢 𝑦 𝐡

508 514.25 727.5 583.25

Consider Univariate ANOVA

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SLIDE 4

C B A 𝑦 𝑑 𝑦 𝐢 𝑦 𝐡

508 514.25 727.5

𝑦 𝐡𝐢𝐷

SIGNAL NOISE 𝐺 = π‘€π‘π‘ π‘—π‘π‘œπ‘‘π‘“ 𝑐𝑓𝑒π‘₯π‘“π‘“π‘œ π‘€π‘π‘ π‘—π‘π‘œπ‘‘π‘“ π‘₯π‘—π‘’β„Žπ‘—π‘œ = 62463.25 672.1943 = πŸ˜πŸ‘. πŸ˜πŸ‘πŸ“πŸ’πŸ˜

583.25

π‘€π‘π‘ π‘—π‘π‘œπ‘‘π‘“ 𝑐𝑓𝑒π‘₯π‘“π‘“π‘œ = 𝑦 𝑗 βˆ’ 𝑦 𝐡𝐢𝐷 2

𝐡,𝐢,𝐷 𝑗

3 βˆ’ 1 βˆ— 4 = 727.5 βˆ’ 583.25 2 + 514.25 βˆ’ 583.25 2 + 508 βˆ’ 583.25 2 2 βˆ— 4 π‘€π‘π‘ π‘—π‘π‘œπ‘‘π‘“ 𝑐𝑓𝑒π‘₯π‘“π‘“π‘œ = πŸ•πŸ‘πŸ“πŸ•πŸ’. πŸ‘πŸ” π‘€π‘π‘ π‘—π‘π‘œπ‘‘π‘“ π‘₯π‘—π‘’β„Žπ‘—π‘œ = 𝑀𝑏𝑠

𝐡 + 𝑀𝑏𝑠 𝐢 + 𝑀𝑏𝑠 𝐷

3 = 891.6667 + 819.3333 + 305.5833 3 π‘€π‘π‘ π‘—π‘π‘œπ‘‘π‘“ π‘₯π‘—π‘’β„Žπ‘—π‘œ = πŸ•πŸ–πŸ‘. πŸπŸ˜πŸ“πŸ’

One-way ANOVA in R:

anova(lm(YIELD~VARIETY))

Used when you have 3 or more samples

Consider Univariate ANOVA

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SLIDE 5

𝐺 = π‘‘π‘—π‘•π‘œπ‘π‘š π‘œπ‘π‘—π‘‘π‘“ 𝐺 = π‘€π‘π‘ π‘—π‘π‘œπ‘‘π‘“ 𝑐𝑓𝑒π‘₯π‘“π‘“π‘œ π‘€π‘π‘ π‘—π‘π‘œπ‘‘π‘“ π‘₯π‘—π‘’β„Žπ‘—π‘œ

π‘€π‘π‘ π‘—π‘π‘œπ‘‘π‘“ 𝑐𝑓𝑒π‘₯π‘“π‘“π‘œ = 𝑦 𝑗 βˆ’ 𝑦 𝐡𝑀𝑀 2

π‘œ 𝑗

π‘œ βˆ’ 1 βˆ— π‘œπ‘π‘π‘žπ‘’ π‘€π‘π‘ π‘—π‘π‘œπ‘‘π‘“ π‘₯π‘—π‘’β„Žπ‘—π‘œ = π‘€π‘π‘ π‘—π‘π‘œπ‘‘π‘“π‘—

π‘œ 𝑗

π‘œ

Probability of observation

π‘‘π‘—π‘•π‘œπ‘π‘š > π‘œπ‘π‘—π‘‘π‘“ π‘‘π‘—π‘•π‘œπ‘π‘š < π‘œπ‘π‘—π‘‘π‘“ P-value

(percentiles, probabilities) Present 1-p-value

In R: pf(F, 𝑒𝑔1, 𝑒𝑔2) In R: qf(p, 𝑒𝑔1, 𝑒𝑔2)

0.50 0.95

∞

∝= 0.05

F-Distribution (family of distributions- shape is dependent on degrees of freedom)

The larger the F-value the further into the tail – AND the smaller the probability that the calculated F- value was found by chance, MEANING there is a high probability that something is causing a significant difference between the groups

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SLIDE 6

Using DISCRIM to predict which group

Problem: A new skull is found but we don’t know whether it belongs to homo erectus or homo habilis or if it’s a new group?

Homo erectus Homo habilis Group centroid New find (unknown origin)

Skull measurement

How predictions work:

  • 1. Calculate group centroid
  • 2. Find out which centroid is the closest position to the unknown data point

New groups are defined when we find a significant difference between new find and predefined groups

Popular method in taxonomy and anthropology

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SLIDE 7

Multivariate Analysis of Variance (MANOVA)

Is there a significant difference among groups based on multiple response variables? (e.g. ANOVA with multiple response variables)

MANOVA in R:

  • utput=manova(responseMatrix~predictorMatrix) (stats package)

Skull measurement

When we calculate a centroid of a group you build a probability distribution around the centroid for comparison You can the run repeated t-tests (with adjusted p-values

for multiple comparisons) to compare the new data to

the groups but MANOVA does it all for you in

  • ne shot!

Another lab on MANOVA for reference: Laura’s website, RENR 480, Lab 22

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SLIDE 8

Assumptions of (MANOVA)

MANOVA is VERY sensitive to invalid assumptions and outliers Within groups we need to have: 1. Normality: Residuals have to be normally distributed 2. Homogeneity of variances: residuals need to have equal variances Need to meet the assumption in the univariate context to meet them for multivariate analyses You therefore first have to check each individual measurement (response variable) for normality and homogeneity e.g. By making boxplots or plotting ANOVA residuals for each variable

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SLIDE 9

Median Mean

Left skewed

negatively skewed

Normal

perfectly symmetric

Right skewed

positively skewed

Represented as a boxplot

Bi-Modal

Two different modes Not necessarily symmetric

Frequency Frequency Mode Mode Mean Median

Assumptions of (MANOVA)

Generate boxplots for each response variable and assess shape & whiskers

Boxplots in R (multiple plots):

boxplot(ResponseVariable~Group)

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SLIDE 10

Testing for Normality & Equal Variances – Residual Plots

Residual plots in R (multiple plots):

plot(lm(ResponseVariable~Group))(2nd plot)

Predicted values Observed (original units) Predicted values Observed (original units) Predicted values Observed (original units) Predicted values Observed (original units)

  • NORMAL distribution: equal number of points along observed
  • EQUAL variances: equal spread on either side of the meanpredicted value=0
  • Good to go!
  • NON-NORMAL distribution: unequal number of points along observed
  • EQUAL variances: equal spread on either side of the meanpredicted value=0
  • Optional to fix
  • NORMAL/NON NORMAL: look at histogram or test
  • UNEQUAL variances: cone shape – away from or towards zero
  • This needs to be fixed for MANOVA (transformations)
  • OUTLIERS: points that deviate from the majority of data points
  • This needs to be fixed for MANOVA (transformations or removal)

Assumptions of (MANOVA)

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SLIDE 11

Assumptions of (MANOVA)

If you violate the assumptions of MANOVA: 1. Transform your data (follow examples we will discuss on the board) 2. Use non-parametric options (e.g. perMANOVA Lab 6)

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SLIDE 12

Multivariate Analysis of Variance (MANOVA) - output

You can see if there is a significant difference across all predictor variables using the Wilk’s MANOVA test statistic Or you can see if there is a significant difference among groups for each predictor variable separately

P-value – the probability the observed difference between groups or larger is due

to random chance Thus if p-value is small this means that something is having an effect on the groups causing the difference