Volledige samenvatting Advanced statistics for Nutritionists

Lecture 1.1 - t-test, confidence interval & sample size
Example:
We want to compare population means of two populations; the population of diabetic
patients following diet A and diet B. The diabetic patients are the units.
We cannot look at all patients, but make a guess about the difference by looking at two
random samples from the populations  the two samples follow from one random sample
of patients and randomisation over the two treatments.
 Experimental study = experiment where treatments can be randomly assigned to
experimental units.
We want to compare the population means μ1 and μ2.
The research hypothesis is that μ1 and μ2 are different; this will be the alternative
hypothesis, so H a : μ1−μ2 ≠ 0. The null hypothesis will be H 0 : μ1−μ2=0.

σ2 = variance = measures the average degree to which number is different from the mean.
The test statistic measures how well the data match up with H0.


ŷ (sample mean) = estimator for μ (population mean)
s (sample standard deviation) = estimator for σ (standard deviation)
s2 (pooled variance estimate) = estimator for σ2 (variance)  the variance of several different
populations with different means for the populations.
Very positive t
suggests μ1−μ 2>0 , so
reject H0.
Very negative t
suggests μ1−μ 2<0 , so
reject H0 too.
But when it’s negative/positive enough?  Rejection Region = the outcomes of t that lead
to rejection of H0.
Degrees of freedom (df) = represent how many values involved in a calculation have the
freedom to vary.
df = (n1- 1) + (n2 – 1)
To determine the rejection region we need to know the distribution of t under H 0, to decide
which values of t are rare.




1

,LET OP! A t- distribution is more flat than a normal distribution.
 If the question concerns the entire population as it is distributed  normal distribution
should be used.
 If question concerns the mean of the population  the t-statistic may be used
P-value = the probability under H0 for the outcome of test statistic t and anything more
extreme (supporting Ha). LET OP! Two sided p-value? 2x the p-value found.
 P-value ≤ α  reject H0, Ha has been shown
 P-value > α  do not reject H0


estimator−value of parameter under H 0
t=
SE
 The estimate is the difference between two sample means
 Value from H0 is often zero
 Standard error is standard deviation of the estimator


One sample t-test
1 random sample, 1 variable  interest in single population mean μ
Example: sample from population of Dutch people
 y = daily salt consumption of a person
 μ = population mean for daily salt consumption of Dutch people
ŷ−6
t= 2
H0: μ = 6, Ha: μ > 6, s
√
n
Paired t-test

1 random sample, 2 variables  interest in difference between population means μ1−μ 2

Example: sample of patients with blood pressure disorder
 y = blood pressure, measured before and after medication
 μ1/2 = population means before and after medication
d−0
t= 2
d = x – y, H0: μd = 0, Ha: μd > 0, s
√
n


Confidence Interval = set of values for which the null hypothesis is accepted  alle
waarschijnlijke waarden voor μ1−μ 2


2

,CI =estimator ± t α (constant ) × SE
2


LET OP! Rejection Region is expressed in t. CI is expressed in the variable you need to know.
Door de α te veranderen, verander je de betrouwbaarheid van het interval. LET OP! Bij een
kleinere confidence interval heb je een hogere betrouwbaarheid nodig  verhoging van de
sample size waardoor de SE kleiner wordt; hoe smaller het interval, hoe nauwkeuriger!
In a (very) large experiment, a difference could be significant, while a narrow interval, and a
small estimate, may tell you that the difference is of no practical importance. Statistically
significant and practically significant is not always the same!
At a given α, for smaller β a
larger sample size n is required.




Power calculation = the probability of correctly rejecting the null hypothesis when it is false
Calculate how big your sample size needs to be:
1) Based on tests; zo veel power nodig voor een significant verschil
 Use a two-independent sample t-test
 Either be negative or positive  two sided alternative hypothesis
 Equation six:
 Suppose probability of 0.95, β = 1 – 0.95
= 0.05
 n = 65? We need at least 65 patients for each
diet, so 130 patients in total.
2) Estimate the difference
 We want a small width of a confidence interval  error margin (E) = half of
the width of the interval.
 When the true value is in the interval, the true value and the estimate will
differ less than E.
 Interested in a confidence interval
 Interval responds to a two-independent samples t-test
 Equation nine:
 N = 77? We need at least 77 patients for each
diet, so 154 in total.
 LET OP! Afronden naar boven; 76.4 wordt 77.




3

, Lecture 1.2 – Analysis of proportions and tables
Example – one proportion & one sample
The proportion of binge drinking among students is 0.44. Let π be the proportion of students
that engage in binge drinking at a particular university. Is π larger than 0.44?
 Experimental units = students
 Response = student is a binge drinker or not (LET OP! Amount doesn’t matter)
Basic observations are binary  1 if student is a binge drinker and 0 if not.
Population mean of binary data is also a population proportion, here the proportion of binge
drinkers in the student population. This is also the probability that a randomly selected
student is a binge drinker  for that reason we use symbol π.
DUS: Gemiddelde van 0’tjes en 1’tjes is in feite ook de proportie.


TEST STATISTIC
H 0 :π =0.44 H a :π >0.44

This is a one-sided alternative hypothesis. The test statistic is number of observed binge
drinkers Y. When Y is too large, we reject H0.
Suppose Y = 240
P−value=P ( y ≥240 ) for π = 0.44

P-value = is de kans dat je testresultaten vindt dat H0 waar is, of extremer.


¿ aantal successen( y )
π=  estimate for π
aantal deelnemers(n)




Calculate 0.95-confidence
interval:




4