Two-Sample Hypothesis Testing in Biostatistics: T-Test and Confidence Intervals, Study notes of Environmental Science

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ESCI 340: Biostatistical Analysis Two-Sample Hypothesis Testing

l_hyp2.pdf (continued) McLaughlin

1 Test for difference between two Means (2-sample t-test) 1.1 Hypotheses: H 0 : μ 1 = μ 2 , alternatively: H 0 : μ 1 − μ 2 = 0, HA: μ 1 ≠ μ 2 HA: μ 1 − μ 2 ≠ 0

1.2 Assumptions 1.2.1 both samples from normal populations 1.2.2 equal population variances

1.3 t

X X

= (^) s (^) X X

−

−

1 2 1 2

• s^ X (^) 1 − X 2 SE of difference betw/ means
• t depends on ν, degrees of freedom

• ν = ν 1 + ν 2 or ν = n 1 + n 2 − 2

• reject H 0 if | t calc| ≥ t α(2),ν ( t α from Zar table B3)

1.4 Calculating s (^) X (^) 1 − X 2

• var (difference between independent variables) = sum of individual variances,

σ

σ σ

X (^) 1 X (^) (^2) n n

2 12 1

22 − (^2)

= + → assume equal variances σ

σ σ

X (^) 1 X (^) (^2) n n

2

2

1

2 − (^2)

= +

• estimate σ^2 ; use both s 12 & s 22 : s

SS SS

p 2 1 2 1 2

=

+

ν +ν

• so: s s n

s X X n

p p 1 2

2 2 1

2 − (^2) = + and s s n

s X X n

p p 1 2

2 1

2 − (^2)

= +

• finally, t X^ X s n

s n

p p

= −

+

1 2 2 1

2 2

1.5 One-Tailed test Hypotheses: H 0 : μ 1 ≥ μ 2 , HA: μ 1 < μ 2 ; if t ≤ − t α ( ), 1 ν then reject H (^0) H 0 : μ 1 < μ 2 , HA: μ 1 > μ 2 ; if t ≥ t α ( ), 1 ν then reject H (^0)

1.6 Violations of Two-Sample t-test Assumptions 1.6.1 normal distribution: t-test is very robust (one-tailed test sensitive to skew)

1.6.2 equal variances: if unequal, greater chance of type I error (greater than α)

correction: Welch’s approximate t

t

X X

s n

s n

'=

−

+

1 2 12 1

22 2

and d.f. ν'=

+

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

−

+

⎛

⎝

⎜

⎞

⎠

⎟

−

s n

s n s n n

s n n

1 2 1

2 2 2

2

12 1

2

1

22 2

2

− often, ν’ not integer; use next smallest integer (e.g., if ν’ = 8.75, use ν = 8)

ESCI 340: Biostat istical Analysis Two-Sample and Paired-Sample Hypothesis Testing 2

l_hyp2.pdf McLaughlin

2 Confidence Limits for Population Means

2.1 1−α Confidence interval for pop i : X t

s i n

p i

±α ( 2 ),ν

2

2.2 If unequal variances, X t s i n i i

±α ( 2 ),ν

2

2.3 Confidence limits for difference betw means: X (^) i − X (^) 2 ± t (^) α ( 2 ),ν sX (^) 1 − X 2 2.4 if H 0 not rejected (samples from pops w/ identical means, μ):

• estimate of μ is weighted average of sample means: X n X^ n X p (^) n n

= +

+

1 1 2 2 1 2

• 1 −α CI for μ: X t

s p n n

p i

±

α ( 2 ),ν +

2 2 3 Testing for Difference betw Variances

3.1 Null Hypothesis (2-tailed): H 0 : σ 1 2 = σ 22 ; HA: σ 1 2 ≠ σ 22

3.2 Variance ratio test F s s

= 1

2 2 2 or^ F^

s s

= 2

2 1 2 ,^ whichever larger; i.e., larger sample variance in numerator 3.3 F -distribution: Zar, Table B

• F α,ν1,ν2 order of ν 1 ,ν 2 matters; F α,numerator df, denominator df

• if H 0 not rejected, best estimate of σ^2 is pooled variance, s p^2

3.4 Variance ratio test severely affected by non-normality

4 Paired-Sample t-test 4.1 Paired data 4.2 Hypotheses: 2-tailed: H 0 : μd = 0, HA: μd ≠ 0

1-tailed: H 0 : μd > μ 0 , HA: μd < μ 0 or: H 0 : μd < μ 0 , HA: μd > μ 0 4.3 Equivalent to a one-sample test d (^) i = difference betw/ paired measurements sample size = # pairs of data degrees freedom = n - 4.4 Test Statistic: t d = (^) sd d = mean difference sd = standard error of mean difference 4.5 Assumption: differences, d i , are normally distributed → not need assumptions from 2-sample t -test (normality & equality of variances)

5 Confidence Limits for Mean Difference 5.1 similar procedure as with one-sample t-test;

5.2 1−α confidence limits for μd : d ± t α ( 2 ), ν sd