random variable distribution. 200 trials where i flipped the coin 50 times and counted heads...

Random variable

Distribution

200 trials where I flipped the coin 50 times and counted heads

His togram (hazenikorunou 52v*200c )

kolik jednicek = 200*1*norm al(x, 24.26, 3.8335)

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

kolik jednicek

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No

of o

bs

no_of_heads in a trial

or I can describe frequencies in percentages

His togram (hazenikorunou 52v*200c )

kolik jednicek = 200*1*norm al(x, 24.26, 3.8335)

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

kolik jednicek

0%

1%

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Per

cent

of o

bs

no_of_ones

or as a cumulative histogramHis togram (hazenikorunou 52v*200c )

kolik jednicek = 200*iNorm al(x, 24.26, 3.8335)

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

kolik jednicek

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no_of_ones

which can also be in percentage scaleHis togram (hazenikorunou 52v*200c )

kolik jednicek = 200*iNorm al(x, 24.26, 3.8335)

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

kolik jednicek

0%

10%

20%

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50%

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80%

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100%

Per

cent

of o

bs

no_of_ones

When my population is infinite

• Then my number of elements is infinite – but I can characterize it as a proportion from all observations in any interval (i.e. probability, that randomly chosen value in be in interval)

• For discrete variable: enumeration of all values and their probabilities pi=P(X=xi) – as a table or formula.

Hus tota pravděpodobnos ti

0 2 4 6 8 10 12 14 16 18 20 22 240.00

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D is tribuč ní funkc e

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1.0

Continuous variable is characterized by distribution function and probability density

probability density Distribution function

Distribution function F(x) =P(X<x) has these basic properties

1. P(a X < b) = F(b) - F(a) ;

2. F(x1) F(x2) pro x1 < x2 ;

3.4.

1)(lim

xFx

0)(lim

xFx

It is actually idealized cumulative histogram with infinitely narrow columns.

How to “idealize” normal histogram

If my columns are endlessly narrow, there will be “nothing” in them – therefore the percentage of observations of the interval is divided by “width” of the column. In a limit case I get for probability density

xxxXxP

xxf

)(

0lim)(

For probability distribution govern:

For distributive function, mean and variance can be computed

Discrete variable

n

i

iipx1

n

i

ii px1

22 )(

Continuous variable

,)( dxxxf

dxxfx )()( 22

Quartile

When this area is 0.75 or 75%

than 12.54 is 75% quantile of distribution (i.e. upper quartile)

Probability distribution

Testing of hypothesis

+ 2 test

I cannot prove any hypothesis

• That’s why I formulate null hypothesis (H0), and with rejection of it I prove its opposite.

• Alternative hypothesis H1 or HA is negation of the null hypothesis

• I, the biologist, am the one who formulates null hypothesis – that’s why null hypothesis would be constructed in such way to be interesting if it is rejected.

Errors in decision

• In the case that the data are random (and it is practically every time in biology) I have to take in account that I can make wrong decision – statistics knows Type I error and Type II error, which are unavoidable part of our decision

• In addition we can make an error by a mistake in computation, but this isn’t necessary :-) .

Recipe for hypothesis’ testing

• 1. I formulate the null hypothesis

• 2. I choose the level of significance and so I obtain critical value (from some tables)

• 3. I compute test criteria from my data

• 4. When the value of test criteria is higher than critical value, I reject the null hypothesis

2 test (test of goodness of fit)

• Example – I hybridize peas: I expect

F1:

F2:

I have 80 offspring – I expect 60:20, I have 70:10

Is it just random variability, or Mendel’s rates doesn’t work in this case?

• 1. Rejecting of null hypothesis about 3:1 ratio is interesting from the biological view. I could test statistically null hypothesis about 4,2371:1 ratio, its rejecting doesn’t bring us any biologically interesting information.

• 2. Null hypothesis will be in the formal way: probability of dominant phenotype’s manifestation is 0.75 (in infinitely large population of potential offspring is ratio of phenotypes 3:1)

Calculation

66.620

)2010(60

)6070( 222

DF=1 (number of categories - 1 for prior given hypothesis), critical value = 3,84

Value of test criteria > critical value, I reject null hypothesis – I say, ratio in F2 is statistically significantly different from expected 3:1 with = 0.05 – or I write (2 = 6.66, df=1, P<0.05)

k

i

k

i i

ii

fff

1

2

1

22

E)EO(

ˆ)ˆ(

f - absolute freqency, i.e. number of random independent observations

What can happen – flipping the coin

Reality – the coin is OK, i.e. P0=P1=0.5 (BUT WE DON’T KNOW THIS) 100 flips, I get 55:45

Than 2=(55-50)2/50+(45-50)2/50 = 1.0 < 3.84.

I cannot reject null hypothesis.

Right decision.


Reality – the coin is OK, i.e. P0=P1=0.5 (BUT WE DON’T KNOW THIS)100 flips, I get 60:40

Then 2=(60-50) 2/50+(40-50) 2/50 = 4.0 > 3.84. I reject null hypothesis on the 5% level of significance. I have made Type I error (and I gibbet innocent). We know the probability of the error: it is . Level of significance is subjected to the probability of rejecting null hypothesis providing that it is true.


Reality – the coin is false, i.e. P0=P1=0.5 (BUT WE DON’T KNOW THIS)

100 flips, I get 60:40

Then 2=(60-50) 2/50+(40-50) 2/50 = 4.0 > 3.84. I reject null hypothesis on 5% level of significance. Right decision (and I gibbet blackguard).

What can happen – flipping the coinReality – the coin is false, i.e. P0=P1=0.5 (BUT WE DON’T KNOW THIS)


Then 2=(55-50)2/50+(45-50)2/50 = 1.0 < 3.84. I cannot reject null hypothesis (and blackguard is free). I have committed Type II error. Its probability is signed as and it is mostly unknown. 1 - is power of the test. Generally, the power of the test is higher with deviation from null hypothesis and with number of observations. As we don’t know , the right formulation of our outcome is: Based on our data we cannot reject null hypothesis. Formulation: We have proved null hypothesis is wrong!

Decision Table Reality H0 is true H0 is false

Our H0 is rejected Type I error Správné rozhodnutí decision H0 isn’t rejected Správné rozhodnutí Type II error

By given number of observations – the better protected against one type error, the more is outcome predisposed to the second one.

If I decide to test on 1% level of significance – the critical value is then 6,63

Right decisionRight decision


Reality – the coin is OK, i.e. P0=P1=0.5 (BUT WE DON’T KNOW THIS)100 flips, I get 60:40

Then 2=(60-50) 2/50+(40-50) 2/50 = 4.0 <6,63. I don’t reject null hypothesis on 1% level of significance. - OK, I didn’t gibbet innocent.


Reality – the coin is false, i.e. P0=P1=0.5 (BUT WE DON’T KNOW THIS)


Then 2=(60-50) 2/50+(40-50) 2/50 = 4.0 < 6,63. I reject null hypothesis on 5% level of significance. Type II error (blackguard is free).

After 20 flips of the coin

head tails chi-squ0 20 201 19 16.22 18 12.83 17 9.84 16 7.25 15 56 14 3.27 13 1.88 12 0.89 11 0.210 10 0

Power of testReality – the coin is false, i.e. P0=P1=0.5 (BUT WE DON’T KNOW THIS) – When it goes exactly according to probability.


Then 2=(55-50)2/50+(45-50)2/50 = 1.0 < 3.84. I don’t reject error II

1000 flips, I get 550:450Then 2=(550-500)2/500+(450-500)2/500 = 10.0 > 3.84. I reject and it is OK.

Reality – the coin is false, i.e. P0=0.51; P1=0.49



1000 hodů, dostávám 510:490


10000 flips, I get 5100:4900Then 2=(5100-5000)2/5000+(4900-5000)2/5000 = 4 > 3.84. I reject and it is OK.

Power of test grows

• With number of independent observations

• With magnitude of deviance from null hypothesis

• With lowering protection against Type I error

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1

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procento

Percentage of heads in a sample sufficient to reject the null hypothesis P1=P2=0.5 by the 2 -test as a function of totalnumber of observations

P>0.05

0.01<P<0.05

P<0.01

P<0.01

percentage

Examples of use

• phenotype ratio

• 3:1

• 9:3:3:1 (number of degrees of freedom = number of categories - 1, for a priori hypothesis, i.e. DF=3)

Examples of use• Sex ratio

• 1:1

• Assumptions!

• Random sampling!

• The same probability

In praxis can be rejecting of null hypothesis sign of three facts:

1. Null hypothesis is wrong.

2. Null hypothesis is right, but the decision os consequence of Type I error.

3. Null hypothesis is right, but the assumptions of the test were violated.

Examples of use• Bee’s orientation

according to the disk colour

• H0: 1:1:1

• How to ensure independence?

• Solid size of sample

Examples of use

• Hardy-Weiberg’s equilibrium

• p2+ 2pq + q

• attention – we take off one degree of freedom more for a parameter that we estimate from data, so DF= 3 - 1 - 1 = 1

What are critical values?

k

i

k

i i

ii

fff

1

2

1

22

E)EO(

ˆ)ˆ(

The higher deviation from null hypothesis, the higher chi-square

What are critical values?

When this is 5%, then 11.1 is critical value on 5% level of significance (here is DF=5)

Nowadays is used more often

We can use the opposite procedure as well. We have computed chi-square=14

The area of the “tail” = P = 0.014 is

“Probability”P is probability, that these or more different result from null hypothesis is just thanks to chance, if H0 is right.

We usually write

• the result is significant on = 0.05 -

• or we write (2 = 6.66, df=1, P<0.05)

And what is about the 2 value is near aroud zero

P>0.99

Can we take it as an evidence of true of H0?

TOO GOOD TO BE TRUE

2 – is deduced just theoretically, butI simulated these values by flipping the coin.

Problem - chi-square is continuous distribution, frequencies are discrete from their definition

That’s why Yates` correlation (on continuity) is sometimes used

k

i i

ii

f

ff

1

22

ˆ

)5.0ˆ(

But this test is too conservative then (i.e. probability of error is usually smaller than α, and so the power of test is smaller too). It is not recommended to use, if the expected frequencies > 5, but isn’t used even if just few of them are smaller.

random variable distribution. 200 trials where i flipped the coin 50 times and counted heads...

Documents

probabilities p i

null hypothesis

coin reality

ones slide

trial slide

right decision slide

power of test reality

prior given hypothesis