Fitting probability models to frequency data. Review - proportions Data: discrete nominal variable with two states (“success” and “failure”) You can do

Fitting probability models to frequency

data

Review - proportions

• Data: discrete nominal variable with two states (“success” and “failure”)

• You can do two things:– Estimate a parameter with confidence interval

– Test a hypothesis

Estimating a proportion

€

ˆ p =x

n

Confidence interval for a proportion*

€

′ p =X + 2

n + 4

€

′ p − Z′ p 1− ′ p ( )

n + 4

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟≤ p ≤ ′ p + Z

′ p 1− ′ p ( )

n + 4

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

where Z = 1.96 for a 95% confidence interval

* The Agresti-Couli method

Hypothesis testing

• Want to know something about a population

• Take a sample from that population• Measure the sample• What would you expect the sample to look like under the null hypothesis?

• Compare the actual sample to this expectation

weird

not so weird

Sample

Test statistic

Null hypothesis

Null distributioncompare

How unusual is this test statistic?

P < 0.05 P > 0.05

Reject Ho Fail to reject Ho

Binomial test

The binomial test uses data to test whether a population

proportion p matches a null expectation for the proportion.

Test statistic

• For the binomial test, the test statistic is the number of successes

Binomial test

The binomial distribution

€

P(x) =n

x

⎛

⎝ ⎜

⎞

⎠ ⎟px 1− p( )

n−x

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Number of correct choices

Frequency

Binomial distribution, n = 20, p = 0.5

€

P(x) =n

x

⎛

⎝ ⎜

⎞

⎠ ⎟px 1− p( )

n−x

x

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


Frequency


x

Test statistic

P-value

• P-value - the probability of obtaining the data* if the null hypothesis were true

*as great or greater difference from the null hypothesis

P-value

• Add up the probabilities from the null distribution

• Start at the test statistic, and go towards the tail

• Multiply by 2 = two tailed test

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


Frequency


x

P = 2*(Pr[16]+Pr[17]+Pr[18]

+Pr[19]+Pr[20])

Sample

Test statistic

Null hypothesis



P < 0.05 P > 0.05


N =20, p0 =0.5

€

P = 2 *(Pr[16] + Pr[17] + Pr[19] + Pr[19] + Pr[20])

= 2

20

16

⎛

⎝ ⎜

⎞

⎠ ⎟ 0.5( )

160.5( )

4+

20

17

⎛

⎝ ⎜

⎞

⎠ ⎟ 0.5( )

170.5( )

3+

20

18

⎛

⎝ ⎜

⎞

⎠ ⎟ 0.5( )

180.5( )

2

+20

19

⎛

⎝ ⎜

⎞

⎠ ⎟ 0.5( )

190.5( )

1+

20

20

⎛

⎝ ⎜

⎞

⎠ ⎟ 0.5( )

200.5( )

0

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟

= 0.0026

This is a pain….

Calculating P-values

• By hand• Use computer software like jmp, excel

• Use tables

Sample

Test statistic

Null hypothesis



P < 0.05 P > 0.05


Discrete distribution

• A probability distribution describing a discrete numerical random variable

Discrete distribution

• A probability distribution describing a discrete numerical random variable

• Examples: – Number of heads from 10 flips of a coin

– Number of flowers in a square meter – Number of disease outbreaks in a year

2 Goodness-of-fit test

Compares counts to a discrete probability

distribution

H0: The data come from a particular discrete probability

distribution.

HA: The data do not come from that distribution.

Hypotheses for 2 test

Test statistic for 2 test

€

2 =Observedi − Expectedi( )

2

Expectediall classes

∑

The day of birth for 350 births in the USin 1999.

Day Sun. Mon. Tues. Wed. Thu. Fri. Sat.Num.of

Births33 41 63 63 47 56 47

H0: The probability of a birth occurring on any given day of

the week is 1/7.

HA: The probability of a birth occurring on any given day of

the week is NOT 1/7.

Hypotheses for day of birth example

Day Sun.

Mon. Tues.

Wed Thu.

Fri.

Sat.

Total

Obs. 33 41 63 63 47 56 47 350

Exp. 50 50 50 50 50 50 50 350

The calculation for Sunday

€

Observed − Expected( )2

Expected=

33− 50( )2

50=

289

50

€


2


∑

=289

50+

81

50+

169

50+

169

50+

9

50+

36

50+

9

50

= 15.24

The sampling distribution of 2 by

simulation

Frequency

2

Sampling distribution of 2 by the 2 distribution

Degrees of freedom

• The number of degrees of freedom specifies which of a family of distributions to use as the sampling distribution

Degrees of freedom for 2 test

df = Number of categories - 1 - (Number of parameters estimated from the data)

Degrees of freedom for day of birth

df = 7 - 1 - 0 = 6

Finding the P-value

Critical value

The value of the test statistic where P = .

12.59

P<0.05, so we can reject the null hypothesis

Babies in the US are not born randomly with respect to the day of the week.

Assumptions of 2 test

• No more than 20% of categories have Expected<5

• No category with Expected 1

2 test as approximation of binomial test

• If the number of data points is large, then a 2 goodness-of-fit test can be used in place of a binomial test.

• See text for an example.

The Poisson distribution

• Another discrete probability distribution

• Describes the number of successes in blocks of time or space, when successes happen independently of each other and occur with equal probability at every point in time or space

Poisson distribution

€

Pr X[ ] =e−μ μ X

X!

Example: Number of goals per side

in World Cup Soccer

Q: Is the outcome of a soccer game (at this level) random?

In other words, is the number of goals per team distributed as expected by pure chance?

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

World Cup 2002 scores

Number of goals for a team

(World Cup 2002)Number of goals Frequency

0 371 472 273 134 25 16 07 08 1

What’s the mean, ?

€

x =37 0( ) + 47 1( ) + 27 2( ) +13 3( ) + 2 4( ) +1 5( ) +1 8( )

128

=161

128

=1.26

Poisson with = 1.26X Pr[X]

0 0.284

1 0.357

2 0.225

3 0.095

4 0.030

5 0.008

6 0.002

7 0

8 0

Finding the ExpectedX Pr[X] Expect

ed

0 0.284 36.3

1 0.357 45.7

2 0.225 28.8

3 0.095 12.1

4 0.030 3.8

5 0.008 1.0

6 0.002 0.2

7 0 0.04

8 0 0.007

} Too small!

Calculating 2

X Expected

Observed

0 36.3 37 0.013

1 45.7 47 0.037

2 28.8 27 0.113

3 12.1 13 0.067

4 5.0 4 0.200

€

Observedi − Expectedi( )2

Expectedi

€


2


∑ = 0.429

Degrees of freedom for poisson




Estimated one parameter,



= 5 - 1 - 1 = 3

Critical valueTable 8.2. An excerpt from the 2 critical value tabl ,e fromAppendi x 3.1

dfX 0.999 0.995 0.99 0.975 0.95 0.05 0.025 0.01 0.005 0.001

1 1.6E-6

3.9E-5 0.00016 0.00098 0.00393 3.84 5.02 6.63 7.88 10.83

2 0 0.01 0.02 0.05 0.1 5.99 7.38 9.21 10.6 13.823 0.02 0.07 0.11 0.22 0.35 7.81 9.35 11.34 12.84 16.274 0.09 0.21 0.3 0.48 0.71 9.49 11.14 13.28 14.86 18.475 0.21 0.41 0.55 0.83 1.15 11.07 12.83 15.09 16.75 20.526 0.38 0.68 0.87 1.24 1.64 12.59 14.45 16.81 18.55 22.467 0.6 0.99 1.24 1.69 2.17 14.07 16.01 18.48 20.28 24.32

Comparing 2 to the critical value

€

2 = 0.429

χ 32 = 7.81

0.49 < 7.81

So we cannot reject the null hypothesis.

There is no evidence that the score of a World Cup Soccer game is not Poisson distributed.

Documents

Fitting probability models to frequency data. Review - proportions Data: discrete nominal variable with two states (“success” and “failure”) You can do