BIOSTATS 4.3 - CH. 24: 1-WAY ANOVA, COMPARING SEVERAL MEANS

• One-way ANOVA compares multiple means to assess if they are significantly different when considered as a group.
• It addresses whether samples could have originated from a single population, rather than focusing on differences between each population mean.

Multiple Comparisons

• The initial step in statistically comparing several populations involves evaluating overall statistical significance.
• This indicates potential differences among the parameters being compared and is determined via the ANOVA F test.
• Subsequent to a statistically significant overall test, a detailed analysis examines pairwise parameter comparisons, to specify parameters that differ.
• Chapter 26 contains more complex methods for detailed analysis of parameter differences

Standard Deviation

• Standard deviation measures the typical size of deviations from the average.

Sample Variance and Standard Deviation

• Sample variance (s²) is calculated using the formula: s² = (1/df) * Σ(xi - x̄)², where df represents degrees of freedom.
• Sample standard deviation (s) is the square root of the sample variance: s = √s².

Population Means

• When sampling from a population with mean µ and standard deviation σ, the variance of sample means approximates the average sample variance.
• Sampling from distinct populations with the same standard deviation σ, but different means µ, results in a greater variance of sample means compared to the average sample variance.

Lab Rat Example

• Male lab rats were assigned randomly into 3 groups with varied food access over 40 days.
• The rat groups were weighed at the end of the experiment in grams.
• Group 1 had access only to chow.
• Group 2 had chow plus access to cafeteria food restricted to 1 hour each day.
• Group 3 had chow plus extended access to cafeteria food all day, every day.
• The experiment tested whether the type of food access influences the body weight of male lab rats.
• H0: µchow = µrestricted = µextended
• Ha: At least one mean µ is different from the others

F Statistic

• The analysis of variance F test compares variation from specific sources with the variation within similar individuals.
• Null hypothesis (H0): All means (µi) are equal.
• Alternative hypothesis (Ha): Not all means (µi) are equal so H0 is false.
• F = (variation among the sample means) / (variation among individuals in the set of samples)

Visualizing F Statistic Size

• Small F statistic variability of means is less than variability within samples,
• Large F statistic variability of means is largger than variability within samples,

Analysis of Variance F Test

• MSG (mean square for groups) represents the variance of means, weighted by sample size, indicating variability of sample averages.
• MSE (mean square for error) or pooled sample variance is the average sample variance, weighted by sample sizes, measuring within-group variability.

Using Table F for Analysis

• The F distribution is asymmetrical and has two degrees of freedom.
• The distribution was discovered by Fisher.
• To use Table F, calculate the F value for the sample data, then find the related area under the curve in Table F.

ANOVA Assumptions

• The k samples must be independent SRSs with unrelated individuals.
• Each population from the k samples should be normally distributed.
• The test is robust to deviations from normality with large-enough samples.
• The ANOVA F-test requires all k populations to have the same standard deviation σ,.
• The ANOVA F test is approximately correct when the largest sample standard deviation is no more than ~ twice as large as the smallest sample standard deviation.
• Equal sample sizes make the ANOVA more robust to standard deviation differences.

ANOVA Conditions & the Lab Rats

• ANOVA requires that the samples are independent and random.
• The rat experiment included a randomized experiment making the 3 groups independent.
• As the sample size is large, the populations should be normally distributed.
• There was no evidence of non-normality in the samples.
• Equal population standard deviations are necessary for ANOVA
• The largest and smallest standard deviations are compared.
• If the largest si is 63.41 and the smallest si is 49.64, the criteria for this condition is met.

Baby Stool ANOVA Example

• Used to check if cigarette contaminants can reach the fetus.
• Meconium, a newborn's stool, indicates exposure to outside contaminants during pregnancy.
• Cotinine, a metabolized form of nicotine, was assessed in 3 newborn samples with different smoking statuses.

Baby Stool Sample

• The experiment checked for equal population standard deviations, determined that largest si can be no more than twice as small as smallest si largest si = 143.67, smallest si = 24.23
• Since 143.67 is more than double 24.23, the ANOVA would not be appropriate here.

ANOVA Calculations

• We have k independent SRSs, from k populations or treatments
• The ith population has a Normal distribution with unknown mean µi.
• All k populations have the same standard deviation σ, unknown
• Formula Η0: μ₁ = μ₂ = ... = μₖ, F = MSG/MSE, F = SSG/(k-1) / SSE/(N-k)

ANOVA Calculations Continued

• Under the null hypothesis (H0), all k samples are from the same population N(μ,σ), sample averages should be no more variable than individual samples' points.
• F has an F distribution with k – 1 (numerator) and N – k (denominator) degrees of freedom if H0 is true.