The document discusses statistical concepts including probability, confidence intervals, hypothesis testing, and inferential statistics concerning means and proportions. It covers methods for comparing paired and unpaired samples, as well as correlations and regression analysis between quantitative variables. Additionally, it outlines key statistical tests such as the chi-squared test and t-tests, emphasizing the importance of sample size and distribution assumptions.

1. FONAMENTS
D’ESTADÍSTICA
Dr. Josep M. Vilaseca
CAPSE 2009

2. PROBABILITY
The probability of an event is the
number of times we would observe it
if we repeated the experiment a large
number of times

3. THE NORMAL DISTRIBUTION
AND THE STANDARD NORMAL
DISTRIBUTION

4. INFERENCE FORM A SAMPLE
MEAN
 The mean of the sampling distribution of
means is the true population mean
 Its standard deviation is the population
standard deviation divided by the square
root of the sample (it is called the standard
error). Mesura la precisió de la meva
mostra.
 Confidence interval: estimated mean +/-
multiplier x standard error of the estimate

5. INFERENCE FORM A SAMPLE
MEAN
 95% confidence interval: we are 95% confident
that the true mean in the population lies between
this interval
 Z and t: using tables we can obtain a probability
for the calculated value
 P-value: is the area under the curve
corresponding to values outside the range (-z,z;
-t,t). That is, the area in the tails of the
distribution gives the probability of observing the
more extreme values

6. INFERENCE FORM A SAMPLE
MEAN
 Null hypothesis: the two population means
are the same
 Alternative hypothesis: the two population
means are not the same
 Hypothesis test: we calculate the
probability of obtaining the observed data
if the null hypothesis were true (“larger
accept, smaller reject”)

7. COMPARISON OF TWO MEANS
 Paired samples occur when the individual
observations in the first sample are
matched to individual observations in the
second sample. For quantitative data this
usually occurs when there are repeated
measurements on the same person
 Unpaired data occur when individual
observations in one sample are
independent of individual observations in
the other

8. COMPARISON OF TWO MEANS
 Paired data: we calculate the difference between
the first and second measurements, then the
mean difference, the standard deviation of the
differences and the standard error of the mean
difference. We can also calculate the probability
that, on average, there is no difference between
the paired observations in the population using a
hypothesis test. The null hypothesis is that the
mean population difference is zero. We assume
that the differences are normally distributed with
a mean of zero

9. COMPARISON OF TWO MEANS
 Unpaired data: we calculate the difference
between two independent means, the standard
deviation in two independent samples, and the
standard error of the difference in two
independent means, which is a combination of the
standard errors of the two independent sample
distributions. Using the standard error of the
difference in means, we can calculate the
confidence interval for the estimated difference
and test whether it is significantly different from
zero. We can use a z test in the same way as we
did before for a single sample mean of paired
samples

10. COMPARISON OF TWO MEANS
 When the sample size is small, we use the
t-distribution to calculate confidence
intervals and test hypothesis (either paired
or unpaired data).
 To compare independent samples,
however, we need to assume that the
variances of the two populations are the
same.

11. INFERENCE FROM A SAMPLE
PROPORTION (7)
 The sampling distribution of a proportion is
approximately Normal when the sample is
large
 The SE of a sample estimate is equal to
the standard deviation divided by √n.
 95% CI= p ± 1.96 x SE(proportion)
 95% CI= p ± 1.96 √p(1 – p) / n

12. INFERENCE FROM A SAMPLE
PROPORTION (7)
 If we want to assess whether the
population proportion has a certain value:
1. First we should state the Null Hypothesis
Π= Π0
2. Then we state the Alternative Hypothesis
Π≠ Π0
3. Finally we compute the test statistic
z= p - Π0 / SE(Π)

13. INFERENCE FROM A SAMPLE
PROPORTION (7)
 Remember: we calculate the SE(Π)
assuming the null hypothesis to be true.
 Remember: these methods are only
reliable if the sample is large (say, if the
proportion is less than 0.5 and the number
of subjects with the disease is 5 or more
 When these conditions are not satisfied,
we use the binomial distribution.

14. COMPARISON OF TWO
PROPORTIONS (8)
 We want to make comparisons between the
proportions in two independent populations
(case – control study, cohort study, clinical
trial).
 For a large sample we can use a normal
approximation to the binomial distribution
 When comparing proportions for
independent samples, the first thing we do
is calculate the difference between the two
proportions

15. COMPARISON OF TWO
PROPORTIONS (8)
 The analysis for comparing two independent
proportions is similar to the comparison of
two independent means
 The standard error for the difference in two
proportions is a combination of the standard
error of the two independent distributions
 Hypothesis test: we use a common
proportion (because the two proportions are
supposed to be the same) and the pooled
standard error

16. ASSOCIATION BETWEEN TWO
CATEGORICAL VARIABLES
 When we want to examine the relationship
between two categorical variables we
tabulate one against the other. This is
called a two – way table (also known as
cross – tabulation)
 An association exists between two
categorical variables if the distribution of
one variable varies according to the value
of the other

17.  The chi – squared test for the 2x2 tables is
identical to the z-test for comparing 2
proportions. The value z is the square root
of chi-squared.
 The Fisher’s exact test may also be used.
ASSOCIATION BETWEEN TWO
CATEGORICAL VARIABLES

18. CORRELATION (10)
 Do the values of a variable tend to be
higher (or lower) for higher values of the
other? CORRELATION
 What is the value of one of the variables
likely to be when we know the value of the
other? LINEAR REGRESSION

19. CORRELATION (10)
 Correlation is used to study the possible linear
(straight line) between two quantitative
variables. This tells how much the two
variables are associated
 To measure the degree of linear association
we calculate a correlation coefficient
 The standard method is to calculate the
Pearson’s correlation coefficient, denoted r

20.  Measures the scatter of the points around
an underlying linear (straight line trend)
 Can take any value from -1 to +1
 If there is no linear relationship then the
correlation is zero. But be careful, there
can be a strong non – linear relationship
between two variables.
CORRELATION (10)
Pearson’s correlation coefficient

21. CORRELATION (10)
 We can think of the square of r as: the
proportion of the variability in the y variable
that is accounted for by the linear
relationship with the y variable
 Assumptions for use of correlation:
 the two variables have an approximately
Normal distribution
 all observations should be independent
 Causation cannot be directly inferred from a
strong correlation coefficient

22. LINEAR REGRESSION (11)
 Regression studies the relationship
between two variables when one of them
depends on the other. This also alows one
variable to be estimated given the value of
the other.

23. MULTI – VARIABLE ANALYSIS 1:
STRATIFICATION (12)
Summary measures
 Standardisation
 Mantel – Haensel

24. MULTI – VARIABLE ANALYSIS 2:
MULTIPLE LINEAR REGRESSION