UWHC Scholarly Forum April 17, 2013 Ismor Fischer, Ph.D. UW Dept of Statistics,

UWHC Scholarly ForumApril 17, 2013

Ismor Fischer, Ph.D.UW Dept of Statistics,UW Dept of Biostatistics and Medical [email protected]

STATISTICS IN A NUTSHELL

UWHC Scholarly ForumApril 17, 2013

Ismor Fischer, Ph.D.UW Dept of Statistics,UW Dept of Biostatistics and Medical [email protected] All slides posted at http://www.stat.wisc.edu/~ifischer/UWHC

http://www.stat.wisc.edu/~ifischer/UWHC

• Click on image for full .pdf article

• Links in article to access datasets

POPULATION

Study Question:Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)?

Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population.

“Statistical Inference”

~ The Normal Distribution ~

symmetric about its mean

unimodal (i.e., one peak), with left and right “tails”

models many (but not all) naturally-occurring systems

useful mathematical properties…

“population mean”

“population standard

deviation”

Example: Body Temp (°F)

low variability

98.6

small

( )f x

Example: Body Temp (°F)

low variability

98.6




deviation”




Example: IQ score

high variability

100

large

( )f x

small




deviation”




Approximately 95% of the population values are contained between

– 2 σ and + 2 σ.

95% is called the confidence level. 5% is called the significance level.

95%2.5% 2.5%≈ 2 σ ≈ 2 σ

“population mean” ( )f x


POPULATION

“Null Hypothesis”

via… “Hypothesis Testing”Study Question:Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)?

H0: pop mean age = 25.4 (i.e., no change since 2010)



cannot be found with 100% certainty, but can be estimated with high confidence (e.g., 95%).

Is the difference STATISTICALLY SIGNIFICANT, at the 5% level?

POPULATION



FORMULA

x1

x4

x3x2 x5

x400

… etc…


sample mean age

1 2 nx x xx

n

Do the data tend to support or refute the null hypothesis?


T-test

?


25.6x

4x5x

2x

3x ?Samples,

size n

1x


… etc…

n

CENTRAL LIMIT THEOREM

Approximately 95% of the sample mean values are contained between

and 2 n 2 n

95%2.5% 2.5%≈ 2 σ ≈ 2 σ



– 2 σ and + 2 σ.

Approximately 95% of the intervals fromto

contain , and approx 5% do not.2x n 2x n

n


contain , and approx 5% do not.2x n 2x n 2

n

2

n

95% margin of error

PROBLEM!σ is unknown the vast majority of the time!



via… “Hypothesis Testing”


sample mean1 2 nx x x

xn

= 25.6

“Statistical Inference”POPULATION

Study Question:Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)?

FORMULA

SAMPLEn = 400 ages

x3x2 x5

x400

… etc…

x1

x4

2n

95% margin of error




POPULATION



FORMULA

SAMPLEn = 400 ages


sample mean1 2 nx x x

xn

= 25.6


x3x2 x5

x400

… etc…

x1

x4

2n

sample standard deviation

sample variance

95% margin of error

= modified average of the squared deviations from the mean


21( )x x 2 21 2( ) ( )x x x x 2 2 21 2( ) ( ) ( )nx x x x x x 2 2 2

2 1 2( ) ( ) ( )

1nx x x x x x

sn

1( )x x

POPULATION



FORMULA

SAMPLEn = 400 ages sample mean

1 2 nx x xx

n

= 25.6


x3x2 x5

x400

… etc…

x1

x4

sample variance

2s s = 1.6 2n

2

s

n

sample standard deviation

= 0.16

95% margin of error


x = 25.6



25.7625.44

BASED ON OUR SAMPLE DATA, the true value of μ today is between 25.44 and 25.76 years, with 95% “confidence” (…akin to “probability”).

x = 25.6

2s

n = 0.16

95% margin of error

2s

n = 0.16

25.7625.44


x = 25.6

95% CONFIDENCE INTERVAL FOR µ

= 25.4

IF H0 is true, then we would expect a random sample mean that is at least 0.2 years away from = 25.4 (as ours was), to occur with probability 1.24%.

x

“P-VALUE” of our sample

Very informally, the p-value of a sample is the probability (hence a number between 0 and 1) that it “agrees” with the null hypothesis. Hence a very small p-value indicates strong evidence against the null hypothesis. The smaller the p-value, the stronger the evidence, and the more “statistically significant” the finding.

Two main ways to conduct a formal hypothesis test:

Very informally, the p-value of a sample is the probability (hence a number between 0 and 1) that it “agrees” with the null hypothesis. Hence a very small p-value indicates strong evidence against the null hypothesis. The smaller the p-value, the stronger the evidence, and the more “statistically significant” the finding.

25.7625.44


x = 25.6

95% CONFIDENCE INTERVAL FOR µ

= 25.4

Two main ways to conduct a formal hypothesis test:

x

“P-VALUE” of our sample

IF H0 is true, then we would expect a random sample mean that is at least 0.2 years away from = 25.4 (as ours was), to occur with probability 1.24%.

FORMAL CONCLUSIONS:

The 95% confidence interval corresponding to our sample mean does not contain the “null value” of the population mean, μ = 25.4 years.

The p-value of our sample, .0124, is less than the predetermined α = .05 significance level.

Based on our sample data, we may (moderately) reject the null hypothesis H0: μ = 25.4 in favor of the two-sided alternative hypothesis HA: μ ≠ 25.4, at the α = .05 significance level.

INTERPRETATION: According to the results of this study, there exists a statistically significant difference between the mean ages at first birth in 2010 (25.4 years old) and today, at the 5% significance level. Moreover, the evidence from the sample data would suggest that the population mean age today is significantly older than in 2010, rather than significantly younger.

POPULATION



FORMULA

x1

x4

x3x2 x5

x400

… etc…


sample mean age

1 2 nx x xx

n

Do the data tend to support or refute the null hypothesis? Is the difference STATISTICALLY SIGNIFICANT, at the 5% level?



25.6x

T-testTwo loose ends

POPULATION






T-test

The reasonableness of the normality assumption is empirically verifiable, and in fact formally testable from the sample data. If violated (e.g., skewed) or inconclusive (e.g., small sample size), then “distribution-free” nonparametric tests can be used instead of the T-test. Examples: Sign Test, Wilcoxon Signed Rank Test (= Mann-Whitney Test)

Two loose ends

Check?

POPULATION



x1

x4

x3x2 x5

x400

… etc…




T-testTwo loose ends

Sample size n partially depends on the power of the test, i.e., the desired probability of correctly rejecting a false null hypothesis. HOWEVER……


contain , and approx 5% do not.2x n 2x nSamples,

size n

1x



– 2 σ and + 2 σ.

4x5x

2x

3x

… etc…


deviation”

95%2.5% 2.5%≈ 2 σ ≈ 2 σ


and 2 n 2 n



contain , and approx 5% do not.2x s n 2x s nSamples,

size n

1x



– 2 s and + 2 s.

4x5x

2x

3x

… etc…


deviation”

95%2.5% 2.5%≈ 2 σ ≈ 2 σ


and 2 s n 2 s n


…IF n is large, 30 traditionally.

But if n is small…

… this “T-score" increases (from ≈ 2 to a max of 12.706 for a 95% confidence level) as n decreases larger margin of error less power to reject.

If n is large, T-score ≈ 2.

If n is small, T-score > 2.

UWHC Scholarly Forum April 17, 2013 Ismor Fischer, Ph.D. UW Dept of Statistics,

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