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inferential_statistics > One-Way ANOVA > Flashcards

One-Way ANOVA Flashcards

1
Q

What method tests significant difference between two independent samples?

A

t-test

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2
Q

What is the formula for determining how many t-tests are required to compare n samples? How many samples are required for 10 samples?

A
  • Number of t-tests with n samples = n! / (2!(n-2)!)
  • 10 samples:
    • 10! / 2! * 8!
    • 10 * 9 / 2
    • 45
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3
Q

For three or more samples, how do you find the distance/variability between means?

A
  • Find the average squared deviation of each sample mean from the total mean.
  • This “total mean” is known as the Grand Mean
  • xbarG
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4
Q

When it comes to the Grand Mean, if the sample sizes are the same for each sample group, how can the Grand Mean be determined?

A
  • If each sample size is the same, then the “mean of means” can be used.
  • For example:
    • Samples X, Y, and Z
    • Each sample is the same size (n)
    • The mean can then be determined by:
      • Adding each average and dividing by the total number of samples (3)
      • (xbar + ybar + zbar) / 3
  • When does this approach not work?
    • When the sizes of the sample are not the same.
    • Then you have to add the average of all samples and divide by the total number of values (sample size of each sample group, added together)
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5
Q
  • What conclusions can we draw from the deviation of each sample mean from the mean of means?
    • What is this known as?
  • What conclusions can we draw from the variability of each sample mean from the mean of means?
    • What is this known as?
A
  • Between-group variability
    • The smaller the distance between sample means, the less likely population means will differ significantly
    • The greater the distance between sample means, the more likely the population means will differ significantly.
  • Within-group variability
    • The greater the variability of each individual sample, the less likely population means will differ significantly
    • The smaller the variability of each individual sample, the more likely population means will differ significantly.
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6
Q

ANOVA

A
  • Analysis of Variance
  • a collection of statisitcal models and their associated procedures (such as “variation” among and between groups) used to analyze the differences among group means.
  • In its simplest form, ANOVA provids a statistical test of whether or not the means of several groups are equal, and therefore generalizes the t-test to more than two groups.
  • ANOVA is useful for comparing (testing) three or more means (groups or variables) for statistical significance.
  • Hypothesis testing:
    • Ho: M1 = M2 = M3
    • HA: At least one pair of samples is significantly different
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7
Q

What does it mean if you get a large statistic during an ANOVA test?

A
  • Two means are causing between subject variability
  • You will reject the null hypothesis (accept the alternative hypothesis)
  • You will need to do an additional step to find out which means are different from each other.
    • These additional tests are called multiple comparison tests
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8
Q

During an ANOVA test, if the variance of a “within-group” individual sample becomes bigger (all else held constant) what does this mean?

A
  • The between sample means are not significantly different
  • We accept the null hypothesis
  • Our test statistic will be smaller because there is a larger within group variability
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9
Q

During an ANOVA test, if the between group variability increases (the sample means get further apart from each other), what does this mean?

A
  • At least one pair of samples is significantly different
  • We accept the alternative hypothesis (reject the null hypothesis)
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10
Q

What is the statistic for the ANOVA test?

A
  • F statistic
  • F = between-group variability / within-group variability
  • Reasoning:
    • Increases in “between-group” variability means accepting the alternative hypothesis
      • Having “between-group” in the numerator will make a large F statistic when there are increases in “between-group” variability
    • Increases in “within-group” variability means accepting the null hypothesis
      • Having “within-group” variability in the denominator means a small F statistic when there are increases in “within-group” variability
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11
Q

What is the formula for ANOVA?

A
  • F = between-group variability / within-group variability
  • F = ( SSbetween / dfbetween ) / ( SSwithin / dfwithin )
  • F = MSbetween / MSwithin
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12
Q

The F-statistic is _________ negative.

A
  • never
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13
Q

SStotal

A
  • SStotal = SSbetween + SSwithin
  • SStotal = sum(xi - xbarG)2
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14
Q

dftotal

A
  • dftotal = dfbetween + dfwithin = N -1
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15
Q

What does the F-distribution look like?

A
  • Righ (positive) skewed
  • Peaks at ‘1’
    • This is due to “no change” in the numerator and “no change” in the denominator being 1 to signify there was not change due to the treatment
  • One critical region in the one tail
    • Critical value and alpha just like t-tests
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16
Q

Clothing Example

  • Given the following datasets, calculate the individual means and the grand mean
    • Snapzi: 15, 12, 14, 11
    • Irisa: 39, 45, 48, 60
    • LolaMoon: 65, 45, 32, 38
A
  • Snapzi
    • xbars = 13
  • Irisa
    • xbarI = 48
  • LolaMoon
    • xbarL = 45
  • Grand Mean
    • xbarG = 35.33
17
Q

Clothing Example

  • Given the following mean values, calculate the SSbetween
    • xbars = 13
    • xbarI = 48
    • xbarL = 45
    • xbarG = 35.33
A
  • SSbetween = n*sum(xbark - xbarG)2
  • SSbetween = 4( (13-35.33)2 + (48-35.33)2 + (45-35.33)2 )
  • SSbetween = 4(498.63 + 160.528 + 93.508)
  • SSbetween = 3010.67
18
Q

Clothing Example

  • Given the following mean and sample values, calculate the SSwithin
    • xbars = 13
      • Snapzi: 15, 12, 14, 11
    • xbarI = 48
      • Irisa: 39, 45, 48, 60
    • xbarL = 45
      • LolaMoon: 65, 45, 32, 38
    • xbarG = 35.33
A
  • Snapzi
    • 15
      • xi - xbars = 15 - 13 = 2
      • (xi - xbars)2 = 4
    • 12
      • xi - xbars = 12 - 13 = -1
      • (xi - xbars)2 = 1
    • 14
      • xi - xbars = 14 - 13 = 1
      • (xi - xbars)2 = 1
    • 11
      • xi - xbars = 11 - 13 = 2
      • (xi - xbars)2 = 4
    • Sum(xi - xbars)2 = 10
  • Repeat for the other two:
    • Sum(xi - xbarI)2 = 234
    • Sum(xi - xbarL)2 = 618
  • Then calculate SSwithin:
    • Sum(xi - xbarK)2 = 10 + 234 + 618 = 862
19
Q

Clothing Example

  • Given the attached image information, calculate the following
    • dfbetween
    • dfwithin
A
  • dfbetween
    • n - 1
    • 3 - 1 = 2
  • dfwithin
    • ​N - K
    • 12 - 3 = 9
20
Q

Clothing Example

  • Given the following data (calculated in the prior examples) calculate the:
    • MSbetween
    • MSwithin
    • F-statistic
  • SSbetween = 3010.67
  • SSwithin = 862
  • dfbetween = 2
  • dfwithin = 9
A
  • MSbetween = SSbetween / dfbetween
    • 3010.67 / 2 = 1505.34
  • MSwithin = SSwithin / dfwithin
    • 862 / 9 = 95.7
  • F-statistic = MSbetween / MSwithin
    • 1505.34 / 95.7 = 15.72
21
Q

Clothing Example

  • Given the following information, calculate the F-critical value and decide whether to accept or reject the null hypothesis:
    • dfbetween = 2
    • dfwithin = 9
    • F-statistic = 15.72
A
  • f-table
    • Find the dfbetween (numerator) along the x-axis = 2
    • Find the dfwithin (denominator) along the y-axis = 9
    • F-critical = 4.2565
  • Accept or reject null hypothesis:
    • Since the F-statistic (15.72) is greater than the F-critical value (4.2565), we reject the null hypothesis in favor of the alternative hypothesis
22
Q

As the variability within treatment groups increases, the likelihood of rejecting the null hypothesis _____________ ?

A
  • decreases
23
Q
  • In ANOVA, the differences between treatment groups (between-group variances) contributes to ____________ ?
A
  • the numerator of the F-ratio
24
Q

What is NOT a potential source of variation within a treatment group?

A
  • Treatment effects
  • Treatment effects would only increase variability between groups, not within a treatment group.
25
Q

As the variability between treatment groups gets larger and larger (assuming the within-group variability reamins relatively constant), the likelihood of rejecting the null hypothesis ____________ ?

A
  • Increases
26
Q

If the null hypothesis is TRUE, then on average the F-ratio for ANOVA is expected to have a value near __________ ?

A
  • 1.00
27
Q

Which combinations of factors is most likely to produce a large F-value?

A
  • Large mean differences, small within-group variability
28
Q

What is a Multiple Comparison Test?

What is an example of a Multiple Comparison Test?

A
  • When an ANVOA test results in the rejection of the null hypothesis, a Multiple Comparison Test helps determine which group means are different
  • Tukey’s Honestly Significant Difference
29
Q

Tukey’s Honestly Significant Difference

A
  • Compares the differences between any two groups means
  • Allows us to make pairwise comparisons
  • THSD = q * sqrt( MSwithin / n)

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