STAT 211 Topic 9

Lecture 18 Notes

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Topic 8 covered comparison of 2 populations based on:

• population mean
• population proportion
• population variable

ANOVA

ANalysis Of VAriance — extension of pooled t-test.

Suppose we have many samples from many populations. Each population has it's own mean… Are all population means equal?

Assumptions:

1. All populations have normal distributions
2. All samples are independent
3. All population variances are equal

We use the F-test due to sums of squares (since $t^{2}=F$)

Procedure

Set up the following hypothesis test:

• H₀: $\mu _{1}=\mu _{2}=\dots =\mu _{I}$
• H_a: H₀ is not true. (at least one μ is different)

1. Sums of Squares

Total

$SS_{tot}=\sum _{i,j}(x_{ij}-{\bar {x}})^{2}$

Treatment

$SS_{trt}=\sum _{i,j}(x_{i}-{\bar {x}})^{2}$

Error

$SS_{trt}=\sum _{i,j}(x_{ij}-{\bar {x}}_{i})^{2}$

Where:

• $x_{ij}$ represents data of j-th sample subject in i-th population (treatment).
• ${\bar {x}}_{i}={\tfrac {1}{J}}\textstyle \sum _{j=1}^{J}x_{ij}$, $J$ represents number of subjects in each sample (${\bar {x}}_{i}$ is the average of each population's sample)
• ${\bar {x}}={\tfrac {1}{IJ}}\textstyle \sum _{i,j}x_{ij}$, $I$ represents number of treatments or populations

2. Mean Squares

We can show that $SS_{tot}=SS_{trt}+SS_{err}$:

Treatment Mean Square

$MS_{trt}={\frac {SS_{trt}}{DF_{trt}}}$, where $DF_{trt}$ is the treatment degrees of freedom: $I-1$

Error Mean Square

$MS_{err}={\frac {SS_{err}}{DF_{err}}}$, where $DF_{err}$ is the error degrees of freedom: $I(J-1)$

Therefore, $DF_{tot}=DF_{trt}+DF_{err}=IJ-1$

3. Test Statistic

Think about the ratio:

$f={\frac {MS_{trt}}{MS_{err}}}\sim F_{DF_{trt},\ DF_{err}}$
(f follows F-distribution with df $DF_{trt}$ and $DF_{err}$, respectively)

If H₀ is true, then f ≈ 1.

4. ANOVA Table

Source	DF	SS	MS	f
Treatment	$I-1$	$SS_{trt}$	$MS_{trt}={\frac {SS_{trt}}{I-1}}$	$f={\frac {MS_{trt}}{MS_{err}}}$
Error	$I(J-1)$	$SS_{err}$	$MS_{err}={\frac {SS_{err}}{I(J-1)}}$
Total	$N-1$	$SS_{tot}$

Perform F-test and reject H₀ if $f>F_{\alpha ,DF_{trt},DF_{err}}$

Example

Study effects of diet pills (four different brands):

1. Randomly assign 20 women to each of 5 different groups: one for each diet pill and one placebo
2. Let women take pills for a month and record weight loss (in pounds):

Group	1	2	3	4	5 (placebo)
Average Loss (${\bar {x}}$)	14	12	10	8	6
Standard Deviation ($s$)	1.3	1.5	0.8	1.0	1.7

If we wanted to see the effectiveness of one brand, do a pooled t-test between that group and group 5 (placebo):

• H₀: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_1 = \mu_2}
• H_a: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_1 > \mu_2}

ANOVA Table:

Source	DF	SS	MS	f
Treatment	4	800	200	118.0638
Error	95	160.93	1.694
Total	99	960.93

If F_{0.05, 4, 95} = 2.47 and F_{0.01, 4, 95} = 3.52, P-value is almost 0 since 118.0638 > 2.47. Therefore, we find enough evidence to conclude that not all of the means are equal.

Multiple Comparison

ANOVA only tells us whether a population mean differs from the others. To find out which ones are different, we could perform multiple t-tests, but that would throw off our significance level.

Tukey's Procedure

Only works if all treatments contain same number of observations.

1. Select α and find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_{\alpha,\, I,\, DF_{err}}} (Q-distribution)
2. Determine Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w = Q_{\alpha,I,\,DF_{err}}\cdot \sqrt{\frac{MS_{err}}{J}}} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J} is the number of observations per treatment.
3. List sample means in increasing order. Underline pairs that differ by less than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w} .
4. Any pairs not underlined by same line are the ones that are significantly different