Lecture 4Statistic for quality

8/20/2019 Lecture 4Statistic for quality

1/21

Mathematical

Biostatistics

Boot Camp:

Lecture 4,

Random

Vectors

Brian Caffo

Table of

contents

Random

vectors

Independence

Independentevents

Independentrandom variables

IID randomvariables

Correlation

Variance and

correlation

properties

Variances

properties of

sample means

The sample

variance

Some

Mathematical Biostatistics Boot Camp: Lecture 4

Vectors

Brian Caffo

Department of Biostatistics

Johns Hopkins Bloomberg School of Public Health

Johns Hopkins University

August 3, 2012


2/21

Mathematical

Biostatistics

Boot Camp:

Lecture 4,

Random

Vectors

Brian Caffo

Table of

contents

Random

vectors

Independence

Independentevents


IID randomvariables

Correlation

Variance and

correlation

properties

Variances

properties of

sample means

The sample

variance

Some

Table of co

1 Table of contents

2 Random vectors

3 IndependenceIndependent eventsIndependent random variablesIID random variables

4 Correlation

5 Variance and correlation properties

6 Variances properties of sample means

7 The sample variance

8 Some discussion


3/21

Mathematical

Biostatistics

Boot Camp:

Lecture 4,

Random

Vectors

Brian Caffo

Table of

contents

Random

vectors

Independence

Independentevents


IID randomvariables

Correlation

Variance and

correlation

properties

Variances

properties of

sample means

The sample

variance

Some

Random v

• Random vectors are simply random variables collected into a v• For example if X and Y are random variables (X , Y ) is a rand

• Joint density f (x , y ) satisfies f > 0 and

f (x , y )dxdy = 1

• For discrete random variables

f (x , y ) = 1

• In this lecture we focus on independent random variables whf (x , y ) = f (x )g (y )


4/21

Mathematical

Biostatistics

Boot Camp:

Lecture 4,

Random

Vectors

Brian Caffo

Table of

contents

Random

vectors

Independence

Independentevents


IID randomvariables

Correlation

Variance and

correlation

properties

Variances

properties of

sample means

The sample

variance

Some

Independent e

• Two events A and B are independent if

P (A ∩ B ) = P (A)P (B )

• Two random variables, X and Y are independent if for any tw

P ([X ∈ A] ∩ [Y ∈ B ]) = P (X ∈ A)P (Y ∈ B )• If A is independent of B then

• Ac is independent of B • A is independent of B c

• Ac is independent of B c


5/21

Mathematical

Biostatistics

Boot Camp:

Lecture 4,

Random

Vectors

Brian Caffo

Table of

contents

Random

vectors

Independence

Independentevents


IID randomvariables

Correlation

Variance and

correlation

properties

Variances

properties of

sample means

The sample

variance

Some

Ex

• What is the probability of getting two consecutive heads?• A = {Head on flip 1} P (A) = .5• B = {Head on flip 2} P (B ) = .5

• A

∩B =

{Head on flips 1 and 2

}• P (A ∩ B ) = P (A)P (B ) = .5 × .5 = .25


6/21

Mathematical

Biostatistics

Boot Camp:

Lecture 4,

Random

Vectors

Brian Caffo

Table of

contents

Random

vectors

Independence

Independentevents


IID randomvariables

Correlation

Variance and

correlation

properties

Variances

properties of

sample means

The sample

variance

Some

Ex

• Volume 309 of Science reports on a physician who was on triatestimony in a criminal trial

• Based on an estimated prevalence of sudden infant death synd8, 543, Dr Meadow testified that that the probability of a mot

children with SIDS was 18,5432

• The mother on trial was convicted of murder• What was Dr Meadow’s mistake(s)?


7/21

Mathematical

Biostatistics

Boot Camp:

Lecture 4,

Random

Vectors

Brian Caffo

Table of

contents

Random

vectors

Independence

Independentevents


IID randomvariables

Correlation

Variance and

correlation

properties

Variances

properties of

sample means

The sample

variance

Some

Example: cont

• For the purposes of this class, the principal mistake was to asprobabilities of having SIDs within a family are independent

• That is, P (A1 ∩ A2) is not necessarily equal to P (A1)P (A2)• Biological processes that have a believed genetic or familiar en

component, of course, tend to be dependent within families

• In addition, the estimated prevalence was obtained from an unon single cases; hence having no information about recurrencefamilies


8/21

Mathematical

Biostatistics

Boot Camp:

Lecture 4,

Random

Vectors

Brian Caffo

Table of

contents

Random

vectors

Independence

Independentevents


IID randomvariables

Correlation

Variance and

correlation

properties

Variances

properties of

sample means

The sample

variance

Some

Usefu

• We will use the following fact extensively in this class:

If a collection of random variables X 1, X 2, . . . , X n are independtheir joint distribution is the product of their individual densit

functions

That is, if f i is the density for random variable X i we have tha

f (x 1, . . . , x n) =n

i =1

f i (x i )


9/21

Mathematical

Biostatistics

Boot Camp:

Lecture 4,

Random

Vectors

Brian Caffo

Table of

contents

Random

vectors

Independence

Independentevents


IID randomvariables

Correlation

Variance and

correlation

properties

Variances

properties of

sample means

The sample

variance

Some

IID random var

• In the instance where f 1 = f 2 = . . . = f n we say that the X i arindependent and identically distributed

• iid random variables are the default model for random sample

• Many of the important theories of statistics are founded on as

variables are iid


10/21

Mathematical

Biostatistics

Boot Camp:

Lecture 4,

Random

Vectors

Brian Caffo

Table of

contents

Random

vectors

Independence

Independentevents


IID randomvariables

Correlation

Variance and

correlation

properties

Variances

properties of

sample means

The sample

variance

Some

Ex

• Suppose that we flip a biased coin with success probability p the join density of the collection of outcomes?

• These random variables are iid with densities p x i (1 − p )1−x i • Therefore

f (x 1, . . . , x n) =

ni =1

p x i (1 − p )1−x i = p x i (1 − p )n


11/21

Mathematical

Biostatistics

Boot Camp:

Lecture 4,

Random

Vectors

Brian Caffo

Table of

contents

Random

vectors

Independence

Independentevents


IID randomvariables

Correlation

Variance and

correlation

properties

Variances

properties of

sample means

The sample

variance

Some

Corre

• The covariance between two random variables X and Y is de

Cov(X , Y ) = E [(X − µx )(Y − µy )] = E [XY ] − E [X• The following are useful facts about covariance

1 Cov(X , Y ) = Cov(Y , X )2 Cov(X , Y ) can be negative or positive3 |Cov(X , Y )| ≤

Var(X )Var(y )


12/21

Mathematical

Biostatistics

Boot Camp:

Lecture 4,

Random

Vectors

Brian Caffo

Table of

contents

Random

vectors

Independence

Independentevents


IID randomvariables

Correlation

Variance and

correlation

properties

Variances

properties of

sample means

The sample

variance

Some

Corre

• The correlation between X and Y isCor(X , Y ) = Cov(X , Y )/

Var(X )Var(y )

1 −1 ≤ Cor(X , Y ) ≤ 12 Cor(X , Y ) =

±1 if and only if X = a + bY for some constant

3 Cor(X , Y ) is unitless4 X and Y are uncorrelated if Cor(X , Y ) = 05 X and Y are more positively correlated, the closer Cor(X , Y ) 6 X and Y are more negatively correlated, the closer Cor(X , Y )


13/21

Mathematical

Biostatistics

Boot Camp:

Lecture 4,

Random

Vectors

Brian Caffo

Table of

contents

Random

vectors

Independence

Independentevents


IID randomvariables

Correlation

Variance and

correlation

properties

Variances

properties of

sample means

The sample

variance

Some

Some useful r

• Let {X i }ni =1 be a collection of random variables• When the

{X i

} are uncorrelated

Var

ni =1

ai X i + b

=

ni =1

a2i Var(X i )

• Otherwise

Var n

i =1ai X i + b

=

ni =1

a2i Var(X i ) + 2n−1i =1

n j =i

ai a j Cov(X i , X

• If the X i are iid with variance σ2 then Var(X̄ ) = σ2/n and E [S


14/21

Mathematical

Biostatistics

Boot Camp:

Lecture 4,

Random

Vectors

Brian Caffo

Table of

contents

Random

vectors

Independence

Independentevents


IID randomvariables

Correlation

Variance and

correlation

properties

Variances

properties of

sample means

The sample

variance

Some

ExampleProve that Var(X + Y ) = Var(X ) + Var(Y ) + 2Cov(X , Y )

Var(X + Y )

= E [(X + Y )(X + Y )] − E [X + Y ]2

= E [X 2 + 2XY + Y 2] − (µx + µy )2

= E [X 2 + 2XY + Y 2] − µ2x − 2µx µy − µ2y

= (E [X 2] − µ2x ) + (E [Y 2] − µ2y ) + 2(E [XY ] − µx

= Var(X ) + Var(Y ) + 2Cov(X , Y )


15/21

Mathematical

Biostatistics

Boot Camp:

Lecture 4,

Random

Vectors

Brian Caffo

Table of

contents

Random

vectors

Independence

Independentevents


IID randomvariables

Correlation

Variance and

correlation

properties

Variances

properties of

sample means

The sample

variance

Some

• A commonly used subcase from these properties is that if a co

random variables {X i } are uncorrelated , then the variance of tsum of the variances

Var

ni =1

X i

=

ni =1

Var(X i )

• Therefore, it is sums of variances that tend to be useful, not sdeviations; that is, the standard deviation of the sum of bunchrandom variables is the square root of the sum of the varianceof the standard deviations


16/21

Mathematical

Biostatistics

Boot Camp:

Lecture 4,

Random

Vectors

Brian Caffo

Table of

contents

Random

vectors

Independence

Independentevents


IID randomvariables

Correlation

Variance and

correlation

properties

Variances

properties of

sample means

The sample

variance

Some

The sample Suppose X i are iid with variance σ

2

Var(X̄ ) = Var1n

n

i =1

X i =

1

n2Var

ni =1

X i

=

1

n2

ni =1

Var

(X i )

= 1

n2 × nσ2

= σ2

n

M h i l


17/21

Mathematical

Biostatistics

Boot Camp:

Lecture 4,

Random

Vectors

Brian Caffo

Table of

contents

Random

vectors

Independence

Independentevents


IID randomvariables

Correlation

Variance and

correlation

properties

Variances

properties of

sample means

The sample

variance

Some

Some com

• When X i are independent with a common variance Var(X̄ ) =• σ/√ n is called the standard error of the sample mean• The standard error of the sample mean is the standard deviat

distribution of the sample mean

• σ is the standard deviation of the distribution of a single obse

• Easy way to remember, the sample mean has to be less variabobservation, therefore its standard deviation is divided by a

√

M th ti l


18/21

Mathematical

Biostatistics

Boot Camp:

Lecture 4,

Random

Vectors

Brian Caffo

Table of

contents

Random

vectors

Independence

Independentevents


IID randomvariables

Correlation

Variance and

correlation

properties

Variances

properties of

sample means

The sample

variance

Some

The sample va

• The sample variance is defined as

S 2 =

ni =1(X i − X̄ )2

n − 1• The sample variance is an estimator of σ2• The numerator has a version that’s quicker for calculation

ni =1

(X i − X̄ )2 =n

i =1

X 2i − nX̄ 2

• The sample variance is (nearly) the mean of the squared deviamean

Mathematical


19/21

Mathematical

Biostatistics

Boot Camp:

Lecture 4,

Random

Vectors

Brian Caffo

Table of

contents

Random

vectors

Independence

Independentevents


IID randomvariables

Correlation

Variance and

correlation

properties

Variances

properties of

sample means

The sample

variance

Some

The sample variance is unb

E n

i =1 (X i −

¯X )

2 =

ni =1

E

X

2

i − nE ¯X 2

=n

i =1

Var(X i ) + µ

2− n Var(X̄

=

ni =1

σ

2

+ µ2− n σ2/n + µ2

= nσ2 + nµ2 − σ2 − nµ2

= (n − 1)σ2

Mathematical


20/21

Mathematical

Biostatistics

Boot Camp:

Lecture 4,

Random

Vectors

Brian Caffo

Table of

contents

Random

vectors

Independence

Independentevents


IID randomvariables

Correlation

Variance and

correlation

properties

Variances

properties of

sample means

The sample

variance

Some

Hoping to avoid some con

• Suppose X i are iid with mean µ and variance σ2• S 2 estimates σ2• The calculation of S 2 involves dividing by n − 1• S /√ n estimates σ/√ n the standard error of the mean• S /√ n is called the sample standard error (of the mean)

Mathematical


21/21

Mathematical

Biostatistics

Boot Camp:

Lecture 4,

Random

Vectors

Brian Caffo

Table of

contents

Random

vectors

Independence

Independentevents


IID randomvariables

Correlation

Variance and

correlation

properties

Variances

properties of

sample means

The sample

variance

Some

Ex

• In a study of 495 organo-lead workers, the following summariefor TBV in cm3

• mean = 1151.281• sum of squared observations = 662361978

• sample sd = (662361978 − 495 × 1151.2812)/494 = 112.62• estimated se of the mean = 112.6215/√ 495 = 5.062

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Lecture 4Statistic for quality