Lecture 1 Interpretation of data

RDP Statistical Methods in Scientific Research - Lecture 1 1

Lecture 1

Interpretation of data

1.1 A study in anorexia nervosa

1.2 Testing the difference between the samples

1.3 Confidence intervals for treatment effects


1.1 A study in anorexia nervosa

Ben-Tovim, Whitehead and Crisp (1979)

“Sufferers from anorexia nervosa, even those whose bodieshave become severely emaciated, often maintain that their bodily dimensions are quite normal”

Are anorexics able to judge their own bodily dimensions?

Are they worse at doing so than healthy controls?

8 anorexics and 11 controls participated in a study of thesequestions


The apparatus

Two lights on a horizontal beam

Move them together: “Say stop when the distance apart is the same as the width ofyour waist”

Repeat while they move apart, and then average the twomeasurements to give the perceived width


Body perception index

Let

A= mean BPI for anorexics

C= mean BPI for controls

Null hypothesis is H0: A= C

perceived widthBPI 100

actual width


The data

Anorexics: 130, 194, 160, 120, 152, 144, 120, 141

Controls: 202, 140, 168, 160, 147, 133, 229, 172, 130, 206, 153

Summary:

Overall (n = 19):mean = 157.95, standard deviation = 30.689

Anorexics (nA = 8):mean = 145.13, standard deviation = 24.398

Controls (nC = 11):mean = 167.27, standard deviation = 32.426


Formulae

mean:

standard deviation (a measure of the spread of the data):

1 nx ... xx

n

2 2

1 nx x ... x xS

n 1


Notes

Here we have means for anorexics and for controls and standard deviations SA for anorexics and SC for controls

These are sample means and sample standard deviations: they vary from sample to sample

The population means are A for anorexics and C for controls and the population standard deviations are A for anorexics and C for controls: these are fixed truths that will never be known precisely

and are estimates of A and C SA and SC are estimates of A and C

Ax Cx

Ax Cx


1.2 Testing the difference between the samples

The two group means are different from one another

Are they significantly different?

Or might the difference just be due to chance?

We will use a t-test to find out


The t-statistic

where

A C

A C

x xt

1 1S

n n

2 2A A C C

A C

n 1 S n 1 SS

n n 2


Notes

We begin with an estimate of the difference between the means:

This is standardised by dividing by S: S2 is a weighted average of

Standardisation ensures that t is unit-free

Division by is a matter of convention,

but it does ensure that values are not too greatly affected by sample sizes

A C

1 1

n n

A Cx x

2 2A CS and S


Calculation

so that

145.13 167.27t 1.622

1 129.377

8 11

2 27 24.398 10 32.426S 29.377

17


Theory

Suppose that

the BPIs of anorexics follow the normal distribution with mean A and standard deviation

the BPIs of controls follow the normal distribution with mean C and the same standard deviation

Then, if A = C, the statistic t follows Student’s t-distributionon 17 degrees of freedom (17 = 19 – 2 = n # parameters)

a similar shape (slightly fatter) centred on 0


The t-distribution

The probability that a random variable following Student’s t-distribution on 17 degrees of freedom is 1.627 is 0.061


Interpretation

If the null hypothesis H0: A= C is true (and the populationshave the same standard deviation), then t is unusually negative

The chance of it being so negative (or even more so) is 0.061

This is the p-value against the one-sided alternative H1: A< C

The value 0.061 is not so small that one would wish to reject H0

and conclude that there is a significant difference – it shows atrend, but does not constitute strong evidence


Caution!

The investigators sought evidence that anorexics had a poorer perception of their bodily dimensions than controls

– that A> C

The trend is in the opposite direction!

“So maybe the anorexics have a better perception, being so obsessed by their bodies”

Investigators are going to wish to interpret the data, whicheverdirection the difference, so use a two-sided p-value


Two-sided p-value

Double the one-sided p-value to give the two-sided p-value:p = 0.122


Convention

A two-sided p-value 0.05 is usually taken to represent strong evidence of an effect

This goes back to Fisher in the 1930s

It is rather arbitrary, but it is a useful yardstick

A one-sided p-value 0.025 is usually taken to represent strong evidence of an effect – this avoids “cheating” by choosing the direction of the difference once the data have been observed


1.3 Confidence intervals for treatment effects

We have used A to denote the population mean of the BPIsfor anorexics and A to denote their population standarddeviations

These are estimated by the sample mean = 145.13 and bythe sample standard deviation SA = 24.398 respectively

How good an estimate of A is 145.13?

How big or small might A actually be?

A confidence interval will answer this question

Ax


Another t-distributed random variable

Let

Note that you cannot calculate tA as it depends on the unknown A

If the BPI observations are normally distributed, then tA followsStudent’s t-distribution with (nA – 1) df

Now, a t7 random variable lies between 2.365 and 2.635 withprobability 0.95

A A AA

A

x nt

S


A confidence interval for A

It follows that, with probability 0.95,

which is

which is

which is

A A A

A

x n2.365 2.365

S

A A A A A A2.365 S n x 2.365 S n

A A A A A A Ax 2.365 S n x 2.365 S n

A145.13 2.365 24.398 8 145.13 2.365 24.398 8


A confidence interval for A

So, with probability 0.95,

We say that (124.73, 165.53) is a 95% confidence interval for A

The upper and lower limits are random, while A is fixed

The limits capture the true value of A with probability 0.95

It could well be that the true mean BPI for anorexics is as low as124.73, it could also be as high as 165.53

A124.73 165.53


A confidence interval for C

For the controls, nC = 11, = 167.27 and SC = 32.426

The 97.5% point of the t distribution on 10 df is 2.228

Hence, the 95% confidence interval for C is

(167.27 2.228 32.426/11) (145.49, 189.05)

Note that the confidence intervals for A and C overlap

What about a 95% confidence interval for = A C?

Cx


A confidence interval for = A C

Now

follows Student’s t-distribution with (n – 2) df

The 97.5% point of the t distribution on 17 df is 2.110

A C

A C

x xt

1 1S

n n


A confidence interval for = A C

It follows that, with probability 0.95

so that the 95% confidence interval for is

A C A CA C A C

1 1 1 1x x 2.110 S x x 2.110 S

n n n n

A CA C

1 1x x 2.110 S

n n

1 122.14 2.110 29.377

8 11

50.94,6.66


Interpretation

0 lies within the confidence interval consistent with lack of significant evidence against

H0: = 0 at the 5% level (2-sided), as found from the t-test

The mean BPI for anorexics could be substantially lower that thatfor controls (by more than 50), or slightly higher

Larger sample sizes would reduce the width of the confidenceintervals, and make it easier to determine whether there really is adifference

Lecture 1 Interpretation of data

Documents

scientific research

mc statistical methods

outstatistical methods

scstatistical methods

mean ma

mean mc

sample standard deviations

bpis of controls