Comparing Means

Comparing Means

Overview of Comparing Means

  • Investigates if a child’s birthweight is affected by the mother's marital status.

  • Analyzes summary measures within the categorical variable of marital status, which is a nominal variable.

Data Presentation

Table 5.16.8: Summary Statistics of Birthweight by Marital Status (Brisbane Babies 1984-1988)

  • Marital Status: The categories involved are Married, Never Married, Living Together, and Other.

  • N: Sample size for each category.

  • Mean (kg): Average birthweight.

  • Standard Deviation (SD) (kg): Variation in birthweights within each category.

  • Minimum (kg): Lowest recorded birthweight.

  • Maximum (kg): Highest recorded birthweight.

Marital Status

N

Mean (kg)

SD (kg)

Minimum (kg)

Maximum (kg)

Married

363

3.57

0.42

2.33

4.85

Never married

70

3.64

0.44

2.34

5.06

Living together

53

3.38

0.42

2.27

4.18

Other

14

3.40

0.40

2.86

4.28

Total

500

3.56

0.43

2.27

5.06

Comparative Analysis

  • Differences in Birthweights:

    • Never Married: Babies are slightly heavier than those born to Married women by 0.07 kg.

    • Living Together: Babies are lighter by 0.19 kg compared to Married women.

    • Other: Babies are lighter by 0.17 kg than Married women.

  • Standard Error of the Mean (SE): Used to assess sampling variability of the mean and derive Confidence Intervals (CIs).

Detailed Birthweight Analysis

Table 5.16.9: Mean Birthweight, SE, and 95% CI by Marital Status

  • Mean: Reflects average birthweights across groups.

  • Standard Error (SE): Estimate of the variance of the sample mean.

  • 95% CI: Ranges that estimate the true population mean based on sample data.

Marital Status

N

Mean (kg)

SE (kg)

95% CI

Married

363

3.57

0.022

(3.53, 3.62)

Never married

70

3.64

0.052

(3.53, 3.74)

Living together

53

3.38

0.058

(3.27, 3.50)

Other

14

3.40

0.108

(3.16, 3.63)

Total

500

3.56

0.019

(3.52, 3.59)

Variability Observations

  • Standard Errors: Differences in group sizes lead to variation in SE:

    • Largest group (‘Married’) has the smallest SE.

    • Smallest group (‘Other’) exhibits the highest SE.

  • Confidence Intervals:

    • Overlap significantly between ‘Married’ and ‘Never married’ groups.

    • ‘Married’ group has a narrow CI that does not overlap with the ‘Living together’ category.

The Two-Sample t-Test

Null Hypothesis Formulation

  • Assumes that true mean birthweights do not vary among different marital status groups.

  • Examines the difference in birthweights:

    • Married: 3.57 kg

    • Never married: 3.64 kg

    • Difference: 0.07 kg.

Test Statistic Dependency

  • Influenced by:

    • Size of the difference between means.

    • Standard errors of the means.

    • Degrees of freedom (df) calculation: df = ext{sum of sample sizes} - 2.

Specific Example of t-Test

  • Comparing birthweights from Married (363) and Never Married (70) mothers.

    • df = 363 + 70 - 2 = 431.

  • Results: p-value is greater than 0.10, indicating lack of significant difference.

Analysis of Variance (ANOVA)

Global Test Analysis

  • Conducts a comprehensive test on four means of marital status with pairwise comparisons examined post hoc.

  • Null hypothesis states no variation in population means across all groups.

ANOVA F-statistic

  • Calculate: F-statistic to evaluate variance among means relative to standard errors of the means.

  • Degrees of Freedom (df) in ANOVA:

    • Numerator df: Number of groups - 1.

    • Denominator df: Total sample size - number of groups.

  • Example Calculation:

    • F = 14.01.

    • Degrees of freedom: df = (3, 497).

    • p-value = 0.0029, indicating significant differences among groups.

Assumptions for t-Test and ANOVA

  • Normal distribution of the mean is critical for small samples.

  • Larger sample sizes suitable for Central Limit Theorem to apply for means distribution.

  • Standard deviations within groups should be approximately equal.

  • Assumed independence among observations from separate groups of individuals.

Non-Normal Distributions: Transformations and Nonparametric Tests

Transformation Approaches

  • Positive skewed distributions can be addressed using log transformation.

  • Non-parametric test alternatives include:

    • For t-test: Wilcoxon–Mann–Whitney test

    • For ANOVA: Kruskal–Wallis test

Repeated Measurements of Quantitative Variable

Contextual Analysis

  • Analyzes whether a variable varies over time, using a dietary intervention trial as an example.

Example Evaluation of Vitamin A Intake

  • Participants: 17 assessed for Vitamin A intake at baseline and 6 months post-intervention.

Table 5.16.10: Vitamin A Levels

Person

Baseline

6 Months

Change (6 months - baseline)

1

120

124

4

2

122

135

13

3

126

140

14

4

131

130

-1

5

135

169

34

9

157

189

32

7

160

176

16

8

181

196

15

9

195

223

28

10

200

254

54

11

205

267

62

12

211

289

78

13

215

301

86

14

220

227

7

15

246

305

59

16

255

278

23

17

278

299

21

Mean (SD)

185.7 (49.6)

217.8 (65.4)

32.1 (26.3)

Assessment of Group Independence

  • Not met; most 6-month values exceed baseline values.

  • Most appropriate summary statistic for repeated measurements: mean change within individuals.

  • Mean Change in Vitamin A Levels: 32.1 micrograms with a range from -1 to +86 micrograms, SD of 26.3 micrograms.

Confidence Interval Calculation

  • Calculate 95% CI for the mean change:

    • ext{95% CI} = ( ext{mean change} - 1.96 imes SE( ext{mean change}), ext{mean change} + 1.96 imes SE( ext{mean change}))

    • Result: (19.5, 44.6), with the lower limit above zero.

Paired or One-Sample t-Test Application

  • Assumes paired measures; evaluates significance of the change within pairs.

  • t-statistic calculation with N pairs (df = N - 1).

  • Calculated t = 5.02, with 16 df and p = 0.0001, indicating statistically significant changes in vitamin A levels.