Introduction to Basic Statistical Methods Part 1: “Statistics in a Nutshell” UWHC Scholarly Forum

Introduction to Basic Statistical Methods

Part 1: “Statistics in a Nutshell”

UWHC Scholarly ForumMarch 19, 2014

Ismor Fischer, Ph.D.UW Dept of [email protected]

STATISTICS IN A NUTSHELL



All slides posted at http://www.stat.wisc.edu/~ifischer/Intro_Stat/UWHC

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

• Right-cick on image for full .pdf article

• Links in article to access datasets

POPULATION“Statistical Inference”

Women in the U.S. who have given birth

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

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

Population Distribution

X


But what does that mean (at least in principle)?

? ?? ?

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

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


X

1x2x

3x

4x 5x


Individual ages from the population tend to collect around a single center with a certain amount of spread, but occasional “outliers” are present in left and right symmetric tails.

More precisely…

ad infinitum…

~ 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: X = Body Temp (°F)

low variability

98.6

small

( )f x

Example: X = Body Temp (°F)

low variability

98.6

Example: X = IQ score

high variability

100


“population mean”


deviation”




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)

“Statistical Inference”

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

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


X

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

POPULATION



FORMULA

x1

x4

x3x2 x5

x400

… etc…


sample mean age1 2 nx x xx

n

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


?


25.6x


T-testT-testT-testT-testX

Actually, this is a special case of…

?Samples,

size n

4x5x

2x

3x1x


… etc…

n

Population Distribution(of ages)

“Sampling Distribution”

(of mean ages)

via mathematical

proof…

X

X

Actually, this is a special case of…

(of ages)


?Samples,

size n

4x5x

2x

3x1x


… etc…

n


(of mean ages)

X

X

CENTRAL LIMIT

THEOREM

… as n gets larger


?


n



(of mean ages)

X

X

The sample mean values have much less variability about than the population values!


(of mean ages)

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.


X

n


and 2 n 2 n

XSample 1

Sample 2 1x

2x

3x

4x

Sample 3

Sample 4

Sample 5

5x

2 n is called the 95% margin of error

etc…

In principle…


and 2 n 2 n

XSample 1

Sample 2 1x

2x

3x

4x

Sample 3

Sample 4

Sample 5

5x


But from the

samples’ point

of view…


and 2 n 2 n

X1x

Sample 1

Sample 2

2x

3x

4x

5x

Sample 3

Sample 4

Sample 5

Approximately 95% of the intervals fromto

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


But from the

samples’ point

of view…


(of mean ages)


and 2 n 2 n

95%2.5% 2.5%≈ 2 σ ≈ 2 σ



– 2 σ and + 2 σ.




X

n



via… “Hypothesis Testing”


sample mean1 2 nx x xx

n

= 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 errorApproximately 95% of the intervals from

to contain , and approx 5% do not.

2x n 2x nPROBLEM!

σ is unknown the vast majority of the time!


POPULATION



FORMULA

SAMPLEn = 400 ages


sample mean1 2 nx x xx

n

= 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

= 1.6 2 sn

2n

1.6


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( ) ( ) ( )1

nx x x x x xsn

1( )x x

POPULATION



FORMULA

SAMPLEn = 400 ages sample mean

1 2 nx x xxn

= 25.6


x3x2 x5

x400

… etc…

x1

x4

sample variance

2s s

sample standard deviation

= 0.16

95% margin of error


400

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

2 sn

= 0.16

95% margin of error

2 sn

= 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 1.28%.

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 (e.g., p < .0001).

Two main ways to conduct a formal hypothesis test:

25.4 25.6

25.4 25.6

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 (e.g., p < .0001).

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.

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 1.28%.

Two main ways to conduct a formal hypothesis test:

x

“P-VALUE” of our sample

However, one problem remains…

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, .0128, 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.

n

(mean ages)


X


and 2 n 2 n

95%2.5% 2.5%≈ 2 σ ≈ 2 σ

Normal Distribution


– 2 σ and + 2 σ.




Normal Distribution

sn

(mean ages)



and 2s n 2s n

95%2.5% 2.5%≈ 2 σ ≈ 2 σ

Normal Distribution


– 2 s and + 2 s.


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


Normal Distribution

T

…IF n is large, e.g., 30

sn

n

X

Alas, this introduces “sampling variability.”

Edited R code:

y = rnorm(400, 0, 1)z = (y - mean(y)) / sd(y)x = 25.6 + 1.6*z

sort(round(x, 1)) [1] 19.6 20.2 20.4 20.5 21.2 22.3 22.3 22.4 22.4 22.4 22.6 22.7 22.7 22.7 22.8 [16] 23.0 23.0 23.1 23.1 23.2 23.2 23.2 23.2 23.2 23.3 23.4 23.4 23.4 23.5 23.5

etc...[391] 28.7 28.7 28.9 29.2 29.3 29.4 29.6 29.7 29.9 30.2

c(mean(x), sd(x))[1] 25.6 1.6

t.test(x, mu = 25.4)

One Sample t-test

data: x t = 2.5, df = 399, p-value = 0.01282alternative hypothesis: true mean is not equal to 25.4 95 percent confidence interval: 25.44273 25.75727 sample estimates:mean of x 25.6

Generates a normally-distributed random sample of 400 age values.

Calculates sample mean and standard deviation.

(mean ages)


(mean ages)



and 2s n 2s n

95%2.5% 2.5%≈ 2 σ ≈ 2 σ

Normal Distribution


– 2 s and + 2 s.


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


Normal Distribution

T

n s

n

n

…IF n is large, e.g., 30

But if n is small…

X

If n is large, T-score ≈ 2.

If n is small, T-score > 2.

… the “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, even if a genuine statistically significant difference exists!

POPULATION



FORMULA

x1

x4

x3x2 x5

x400

… etc…


sample mean age1 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 (80% or more).

Introduction to Basic Statistical Methods

Part 1: Statistics in a Nutshell



Part 2: Overview of Biostatistics: “Which Test Do I Use??” Sincere thanks to…

• Judith Payne

• Heidi Miller

• Samantha Goodrich

• Troy Lawrence

• YOU!

Documents

Introduction to Basic Statistical Methods Part 1: “Statistics in a Nutshell” UWHC Scholarly Forum