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Medical Statistic
What is a statistic
•
A statistic is a field of study concerned with methods of
Collection‫ تجميع‬,Organization‫تنظيم‬, Summarizing‫ تلخيص‬,
• It makes presentation ‫عرض‬, Classification‫وصف‬, and
analyzing ‫ تحليل‬the data. This analysis may lead to conclusions
‫نتائج‬and decision ‫ قرارات‬.
• A statistician )‫ (االحصائي‬is someone who is versed ‫متمكن‬in the
successful ways application of statistical analysis.
Medical Statistics
deals with applications of statistics to medicine
and the health sciences, including
epidemiology)‫(علم االوبئة‬,
public health)‫(الصحة العامة‬,
forensic medicine)‫(الطب العدلي‬,
clinical research)‫ (البحوث السريرية‬.
Types of Statistics Science:
statistic may be: (i) Descriptive ; (ii) Inferential
• When we first collect data for some project, it will usually
be in a ‘raw ‫ ’خام‬form . That is, not organized ‫غير منظمه‬in
any way, making it difficult to see what’s going on.
Descriptive statistics is a series of procedures designed to
illuminate ‫ القاء الضوء‬the data, This may mean sorting the
data by size; perhaps putting it into a table, maybe
presenting it in an appropriate chart, or summarizing it
numerically; and so on.
• An important consideration in this process is the type of
variable concerned. The data from some variables are
best described with a table, some with a chart, some,
perhaps, with both. Other variables, a numeric summary
is more appropriate.
• When we want to discover‫ اكتشاف‬things that
interest us about a population, we take a
sample. We then hope to generalize our
sample findings, Statisticians call this process,
of generalizing from a sample to a population,
statistical inference .
It is include two main parts:
• Estimation ‫التقدير‬
• Test of Hypothesis ‫اختبار الفرضيات‬
Population & Sample
• Population: This is a much larger group that
containing all the values & observations
about which we wish to make inference.
• Sample: This is a part of population generally
selected so as to be representative of the
population in the variable(s) under study.
Variables and data
• A variable is something whose value can vary.
For example, age, sex and blood type are
variables.
• Data are the values you get when you
measure a variable. For example, 32 years(for
the variable age), or female (for the variable
sex).
Types of Data
• Categorical (Qualitative)
Nominal (no natural ordering)
Blood groups: A,B,O,AB, Sex :m/f
Ordered categorical (ordinal)
Pain severity - mild, moderate, severe
Grade of breast cancer, Social Class: I to V
• Numerical (Quantitative)
Discrete
Number of children in a family, Number of attacks of
asthma ‫نوبات الربو‬per week
Continuous
Age (in years, days, hours, seconds….)
Height in cm , Weight in kg
‫ء‬
Statistical Notations
The information in statistic may be a number of variables (say X, Y, Z…etc.)
and each variable is a number observation (say Xi, Yi, Zi… etc.), then:
Rules of the Summation:
• ∑c = n.c
(c is a constant)
• ∑(c yi) = c ∑yi
• ∑(xi±yi) = ∑xi±∑yi
• ∑ (xi± c) = ∑xi±n.c
• ∑(xi)2 ≠(∑xi)2
• ∑(xi / yi) ≠∑xi / ∑yi
• ∑(xiyi) ≠∑xi ∑yi
•
• Example:
• Calculate 1to 7 above if:
• C=6 , Xi: 3, 5, 1 , Yi: 4, 2, 10
Table(1): Systolic blood pressure levels in 16 middle
aged men before and after a standard exercise
Displaying and Summarizing Data
Display Numerical Data
Dot plot
Stem & leaf plot
Histogram
Box & whisker plot
For our example Systolic blood pressure-Before-
Displaying and Summarizing Data
Display Categorical Data
Bar Chart
Pie
measures of location
A summary measure of location is a value around which most of the data values tend to congregate or
center. The most famous three measures of location we are discussed are : the mode; the median ;
and the mean. The choice of appropriate measure to be used; depends on the type of data
• The mode
The mode is that value in the data that has the highest frequency (i.e. occurs the most often).The mode
is not useful with metric continuous data where no two values may be the same. The other
shortcoming of this measure is that there may be more than one mode in a set of data.
• The median
If we arrange the data in ascending order of size, the median is the middle value. Thus, half of the
values will be equal to or less than the median value, and half equal to or above it.
The median is thus a measure of central-ness. As an example of the calculation of the median, suppose
you had the following data on age (in ascending order of years), for five individuals:30 ,31, 32, 33,
35. The middle value is 32, so the median age for these five people is 32 years. If you have an even
number of values, the median is the average of the two values either side of the ‘middle’. If you
have n values arranged in ascending order, then : the median = 1/2(n + 1)th value.
An advantage of the median is that it is not much affected by skewness in the distribution, or by the
presence of outliers. However, it discards a lot of information, because it ignores most of the
values, apart from those in the center of the distribution.
• The mean
The mean, or the arithmetic mean (the average) . One advantage of the mean over the median is that it
uses all of the information in the data set. However, it is affected by skewness in the distribution, In
other hand, it cannot be used with ordinal data (recall that ordinal data are not real numbers, so
they cannot be added or divided).
measures of location
Pros and cons of mean/median/mode
• Median robust to outliers.
•
•
•
•
•
Median/mode reflects what ‘most’ people experience.
Mean uses all the data (more ‘efficient’).
Mean is ‘expected’ value.
Mean more common with statistical tests.
Mode useful for grouped or categorical data
A guide to choosing an appropriate measure of location
Type of variable
mode
median
mean
Qualitative - Nominal
yes
No
No
Qualitative - Ordinal
yes
yes
No
Quantitative - Discrete
yes
Yes
yes
Quantitative - Continuous
No
Yes
yes
with Reservedly
measures of spread
• The dispersion is the difference of each values in the
observations from the arithmetic mean. If all values are the
same then there is no dispersion, and the dispersion could be
small (ex. Sample of male & female with weights between 69
– 79), or large (ex. weights between 90 – 120).
• There are two types of dispersion measures
Absolute Variation Measures:
Range. Variance. Standard Deviation . Standard Error of the Mean.
Relative Variation Measure:
Coefficient of Variation.
•
•
•
•
Range  X l arg est  X smallest
The Range :
X  X 
The Variance : S   n  1
,
The Standard Deviation: S.D.  S 
The Standard Error of the Mean:
2
2
i
2
S2
SE 
SD
n
The SD measures the variability of each observation from the mean of the sample , The SE give
idea from How much the sample mean is far away from the population mean
• The Coefficient of Variation (CV):
CV 
SD
*100
X
The CV is used to compare the dispersion in two sets of data even when the units are different .
Choosing an appropriate measure of spread
Type of variable
Type of measure
Range
Variance & Stander Deviation
Qualitative - Nominal
No
No
Qualitative - Ordinal
Yes
No
Quantitative - Discrete
yes
Yes
New from table(1) Systolic blood pressure-Before-
Or by using a second formula
Since V= 185.58 , SD= 13.6 , SE= 3.4
Systolic blood pressure-After- : Homework
• Interpretation of results
• If you have X  4 , S x2  0.23 , a  3 , b  5 and Y= X + a , Z = aX-b , then estimate:
Y , S y2 , Z , S z2
Inference statistical
•
•
•
•
•
•
As you remember :
Population :All individual s in which we are interested in.
Sample :Group of individuals drawn from our population of interest which we
study in order to learn about the population
Sample statistics and population parameters
Population Parameter
Sample Statistic
Mean
µ
X
SD
σ
S
The main aim of statistical analysis is to use the information gained from a sample
of individuals to make inferences about the population of interest.
There are two basic approaches to statistical analysis:
Hypothesis testing & Estimation.
Main Steps Of Hypothesis Testing:
• 1. State your null hypothesis (H0) & (H1), (H0) : (Statement you are looking
for evidence to disprove).
• 2. Choose a significance level, α, for the test.
• 3. Chose a suitable test and calculate statistic.
• 4. Compare between tabulated and calculated statistic o make a decision
about whether to reject, or not reject, your null hypothesis.
Hypothesis Testing Procedures
Parametric Test
3- The ANOVA Model : Test of hypothesis about differences among three or more population means
• Source of variance in easily formula
ANOVA Table
Source of
Variation
(S.O.V)
Between
Treatments
Within
Treatments
Total
Degree of
Sum of Squares Mean Squares
Calculated F
Freedom (d.f.)
(S.S.)
(M.S.)
k-1
SSk
MST
MST/MSE
N-k
SSe
N-1
SST
Tabulated F
MSE
Fα((t-1),(N-t))
Examples
Q1- Return to Systolic blood pressure levels data, (use tab.t(0.05,15)=2.13
• 1) Test if the mean of Systolic blood pressure levels is 148(mmHg)in Before data.
• 2)Test if their is a significant different between Before exercise & After exercise
Solution
1) H0 :µ=148 Vs H1 :µ≠148
Since we have one sample with n< 30, we used t-test which
y  141.1,   0, s  13.6, n  16
cal.t  41.5  tab.t  2.13
there is a significant different in Before data
2) H0 :µA -µB=0
µd = 0
Vs H1 : µA -µB≠0
µd ≠ 0
2) Since we have paired observation Before exercise & After exercise ,we used t-test
which t  d S 
0
d
n
d = 6.63 , o  0, Sd  5.97, n  16
cal.t  4.45  tab.t  2.13
there is a significant different
2- A paper manufacturer makes grocery bags. They are interested in
increasing the tensile strength of their product. It is thought that strength is a
function of the hardwood concentration in the pulp. An investigation is
carried out to compare four levels of hardwood concentration: 5%, 10%, 15%
and 20%. Six test specimens are made at each level and all 24 specimens are
then tested in random order. The results are shown below
.
Exercises
1- A study reported the mean birth weight of 98 infants who were born prematurely as 1.31 kg
with σ = 0.42 kg, test if the mean of birth weights population = 1.5 ,(use 1.96 as a tabulated
value)
2- An outbreak of Salmonella-related illness was attributed to ice cream produced at a certain
factory. Scientists measured the level of Salmonella in 9 randomly sampled batches of ice
cream. The levels (in MPN/g) were: 0.593, 0.142, 0.329, 0.691, 0.231, 0.793, 0.519, 0.392,
0.418 . Is there evidence that the mean level of Salmonella in the ice cream is greater than
0.3 MPN/g? ,(use 2.306 as a tabulated value)
3- 6 subjects were given a drug (treatment group) and an additional 6 subjects a placebo
(control group). Their reaction time to a stimulus was measured (in ms). We want to perform
a two-sample t-test for comparing the means of the treatment and control groups. Control
Group:(91, 87, 99, 77, 88, 91) , Treat. Group:(101, 110, 103, 93, 99, 104), ,(use 2.228 as a
tabulated value)
Exercises