Two-Sample Variances

Introduction

  • In the last module, we reviewed statistical inference for one sample.

  • In this module, we will focus on tests for two samples.

    • Two-sample independent means (\mu_1-\mu_2)
    • Two-sample dependent means (\mu_d)
    • Two-sample proportions (\pi_1-\pi_2)
    • Two-sample variances (\sigma^2_1/\sigma^2_2)

  • In this lecture: two-sample variances (\sigma^2_1/\sigma^2_2)

R Setup

library(tidyverse)
library(ssstats)
  • Recall our Zootopia data,
zootopia <- read_csv("https://raw.githubusercontent.com/samanthaseals/SDSI/refs/heads/main/files/data/lectures/1-zootopia.csv")
resident_name department district academy_status academy_exam_score sleep_hours hustle_earnings district_temp_f
Sable Jones Desk Rainforest District Fail 65 5.5 14.72 72.2
Kevin Frostwhisker Patrol Downtown Fail 69 7.4 45.66 66.2
Jasper Duskrunner Patrol Sahara Square Pass 79 8.7 30.91 89.8

Confidence Interval for \sigma^2_1/\sigma^2_2

  • (1-\alpha)100\% CI for two population variances, \sigma^2_1/\sigma_2^2

\left(\dfrac{s_1^2}{s_2^2}\cdot\dfrac{1}{F_{\alpha/2,\,n_1-1,\,n_2-1}}, \ \ \dfrac{s_1^2}{s_2^2}\cdot F_{\alpha/2,\,n_2-1,\,n_1-1}\right)

Confidence Interval for \sigma_1/\sigma_2

  • (1-\alpha)100\% CI for two population variances, \sigma_1/\sigma_2

\left(\sqrt{\dfrac{s_1^2}{s_2^2}\cdot\dfrac{1}{F_{\alpha/2,\,n_1-1,\,n_2-1}}}, \ \ \sqrt{\dfrac{s_1^2}{s_2^2}\cdot F_{\alpha/2,\,n_2-1,\,n_1-1}}\right)

Hypothesis Test for \sigma_1^2/\sigma_2^2

  • Hypotheses:

Two-Tailed

H_0: \ \sigma_1^2 = \sigma_2^2

H_1: \ \sigma_1^2 \ne \sigma_2^2

Left-Tailed

H_0: \ \sigma_1^2 \ge \sigma_2^2

H_1: \ \sigma_1^2 < \sigma_2^2

Right-Tailed

H_0: \ \sigma_1^2 \le \sigma_2^2

H_1: \ \sigma_1^2 > \sigma_2^2

  • \sigma_i^2 is the population variance for group i

Statistical Inference on Two Variances (R)

  • We will use the two_variances() function from library(ssstats) to perform statistical inference.

  • Generic syntax:

dataset_name |> two_variances(outcome = continuous_variable,
                              grouping = grouping_variable,
                              first = "first_group_in_ratio",
                              alternative = "two" | "less" | "greater",
                              alpha = specified_alpha)
1
The continuous outcome column.
2
The grouping column that splits your data into the two independent samples being compared. Must have exactly two levels.
3
The group that goes in the numerator of the ratio (\sigma^2_{first}/\sigma^2_{other}).
4
The direction of H_1; default is "two".
5
The significance level \alpha, which also sets the CI’s confidence level at (1-\alpha)100\%; default is 0.05.

Example 1

  • Deputy Chief is concerned that patrol officers experience less variability in their side hustle incomes.

  • We will use the zootopia dataset to answer the following question:

    • Is there evidence that side-hustle income variability is smaller for patrol officers?

  • We’ll work with hustle_earnings and department.

Example 1

  • Point and interval estimates:
zootopia |> two_variances(outcome = hustle_earnings, 
                          grouping = department, 
                          first = "Patrol")
Two-sample variances

s²[Patrol] = 346.664 (s = 18.6189)
s²[Desk] = 785.4795 (s = 28.0264)
Ratio: s²[Patrol] / s²[Desk] = 0.4413

95% CI for σ²[Patrol] / σ²[Desk]: (0.1345, 1.2665)

Hypotheses:
H₀: σ²[Patrol] / σ²[Desk] = 1
H₁: σ²[Patrol] / σ²[Desk] ≠ 1
Test statistic: F(17, 11) = 0.441
p-value: 0.126
Conclusion: Fail to reject the null hypothesis (p = 0.126 ≥ α = 0.050)

Example 1

  • Is there evidence that side-hustle income variability is smaller for patrol officers?

  • How do we translate this into hypotheses?

    • “variability is smaller for patrol” \to directional!

  • Hypotheses:

    • H_0: \sigma^2_{\text{Patrol}} \ge \sigma^2_{\text{Desk}}
    • H_1: \sigma^2_{\text{Patrol}} < \sigma^2_{\text{Desk}}

  • For the two_variances() function, we will set

    • first = "Patrol"
    • alternative = "less".

Example 1

  • Is there evidence that side-hustle income variability is smaller for patrol officers?
zootopia |> two_variances(outcome = hustle_earnings, 
                          grouping = department, 
                          first = "Patrol",
                          alternative = "less")
Two-sample variances

s²[Patrol] = 346.664 (s = 18.6189)
s²[Desk] = 785.4795 (s = 28.0264)
Ratio: s²[Patrol] / s²[Desk] = 0.4413

95% CI for σ²[Patrol] / σ²[Desk]: (0.1345, 1.2665)

Hypotheses:
H₀: σ²[Patrol] / σ²[Desk] ≥ 1
H₁: σ²[Patrol] / σ²[Desk] < 1
Test statistic: F(17, 11) = 0.441
p-value: 0.063
Conclusion: Fail to reject the null hypothesis (p = 0.063 ≥ α = 0.050)

Example 1

  • Hypotheses

    • H_0: \sigma^2_{\text{Patrol}} \ge \sigma^2_{\text{Desk}}
    • H_1: \sigma^2_{\text{Patrol}} < \sigma^2_{\text{Desk}}

  • Test Statistic and p-Value

    • F_0 = 0.441
    • p = 0.063

  • Rejection Region

    • Reject H_0 if p < \alpha; \alpha = 0.05

  • Conclusion and Interpretation

    • Fail to reject H_0. There is not sufficient evidence to suggest that the variability in side ustle earnings differs between those on patrol or desk duty.

Example 2

  • Today, Deputy Chief wants to look into the amount of sleep between the two departments. In particular, Deputy chief believes that patrol’s sleep habits are less predictable than those on desk duty.

  • We will use the zootopia dataset to answer the following question:

    • Is there evidence that Patrol officers’ sleep hours are more variable than Desk officers’?

  • We’ll work with sleep_hours and department.

Example 2

  • Point and interval estimates:
zootopia |> two_variances(outcome = sleep_hours, 
                          grouping = department, 
                          first = "Patrol")
Two-sample variances

s²[Patrol] = 1.5579 (s = 1.2482)
s²[Desk] = 2.0242 (s = 1.4228)
Ratio: s²[Patrol] / s²[Desk] = 0.7696

95% CI for σ²[Patrol] / σ²[Desk]: (0.2345, 2.2085)

Hypotheses:
H₀: σ²[Patrol] / σ²[Desk] = 1
H₁: σ²[Patrol] / σ²[Desk] ≠ 1
Test statistic: F(17, 11) = 0.77
p-value: 0.608
Conclusion: Fail to reject the null hypothesis (p = 0.608 ≥ α = 0.050)

Example 2

  • Is there evidence that Patrol officers’ sleep hours are more variable than Desk officers’?

  • How do we translate this into hypotheses?

    • “variability is higher for patrol” \to directional!

  • Hypotheses:

    • H_0: \sigma^2_{\text{Patrol}} \le \sigma^2_{\text{Desk}}
    • H_1: \sigma^2_{\text{Patrol}} > \sigma^2_{\text{Desk}}

  • For the two_variances() function, we will set

    • first = "Patrol"
    • alternative = "greater".

Example 2

  • Is there evidence that Patrol officers’ sleep hours are more variable than Desk officers’?
zootopia |> two_variances(outcome = sleep_hours, 
                          grouping = department, 
                          first = "Patrol",
                          alternative = "less")
Two-sample variances

s²[Patrol] = 1.5579 (s = 1.2482)
s²[Desk] = 2.0242 (s = 1.4228)
Ratio: s²[Patrol] / s²[Desk] = 0.7696

95% CI for σ²[Patrol] / σ²[Desk]: (0.2345, 2.2085)

Hypotheses:
H₀: σ²[Patrol] / σ²[Desk] ≥ 1
H₁: σ²[Patrol] / σ²[Desk] < 1
Test statistic: F(17, 11) = 0.77
p-value: 0.304
Conclusion: Fail to reject the null hypothesis (p = 0.304 ≥ α = 0.050)

Example 2

  • Hypotheses

    • H_0: \sigma^2_{\text{Patrol}} \ge \sigma^2_{\text{Desk}}
    • H_1: \sigma^2_{\text{Patrol}} < \sigma^2_{\text{Desk}}

  • Test Statistic and p-Value

    • F_0 = 0.77
    • p = 0.304

  • Rejection Region

    • Reject H_0 if p < \alpha; \alpha = 0.05

  • Conclusion and Interpretation

    • Fail to reject H_0. There is not sufficient evidence to suggest that the variability in side ustle earnings differs between those on patrol or desk duty.

Wrap Up

  • In this lecture, we discussed statistical inference on two variances or standard deviations

  • In this module, we have covered two-sample tests.

  • In the next module, we will learn one-way ANOVA.