Добавил:
Sekretar
kiopkiopkiop18@yandex.ru
t.me/Prokururor I Вовсе не секретарь, но почту проверяю
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_155_библиотеки_им_акад_М_И_Перельмана
.pdf
11.16 Exercises 269
methods, because both methods (confidence interval and p-value) are based on the
same theoretical distribution of the test statistic.
Thus, when p <α shows that the data is inconsistent with the null hypothesis
and the null hypothesis should be rejected, the confidence interval of
% 100
(
1 − α
)
will not include the value specified in the null hypothesis and indicates rejecting
the null hypothesis. When
p >α, shows the compatibility of the data with the null
hypothesis that there is no reason to reject the null hypothesis, the confidence interval
of
% 100
(
1 − α
)
will also include the value specified in the null hypothesis and the
null hypothesis cannot be rejected. For example, if the significance level is considered
at 0.05 and the value is p < 0.05, the null hypothesis is rejected at the confidence
level of 95%, and the 95% confidence interval will not include the specified value in
the null hypothesis, which is an evidence to reject the null hypothesis.
The confidence interval shows uncertainty and inaccuracy in estimating the
studied variable. Consequently, it has more and more useful information than p.
However, if the confidence interval is presented alone, only the experienced
statistician can estimate the approximate value of p. Therefore, it is desirable to
present both p and confidence intervals in scientific articles and reports.
11.15 How to Present the Results of a Statistical Test
The results of the research in scientific papers and reports should provide the exact
p-value in addition to the confidence interval for the means and percentages and
avoid stating the limits for p-value as a range (p < 0.05). In cases where the test is
meaningless, avoid using phrases such as “p = NS”. Because by providing the exact
p-value (e.g., p = 0.092), the reader will have a clearer interpretation of the results of
your research. Only in cases where the p-value is obtained by the researcher himself
from the statistical tables (not by software), he can express the p-value between the
two given values in the table (e.g., 0.05 < p < 0.01). Usually, it is enough to provide
p-value up to two decimal places after zeros (e.g., p = 0.14 or p = 0.0034). Only in
cases where the p-value is too small, its presentation is reported as p < 0.001.
11.16 Exercises
1. The mean and standard deviation of pain relief time of a type of reliever were
announced 25 min and 18 min after consumption. A r esearcher bought a s ample
of 10 of those painkillers randomly from the city and gave them to patients. The
mean pain relief time in this sample was 32 min. Does the researcher’s sample
support the factory claim based on the time of pain relief of these pills?
https://t.me/med1917
270 11 Hypothesis Testing
Table 11.14 Shoe size of women by hospital wards
Shoe size 36 37 37.5 38 38.5 39 40+ To t a l
Cesarean ward 5 7 6 7 8 10 43 86
Other wards 17 28 36 41 46 140 308 616
2. An entomological researcher has applied a statistical model to investigate the
effect of aphids on cotton bolls (buds). He counted the number of aphids on
the 20 cotton blossoms on a particular day, which are: 42–26–15–19–27–32–
26–24–17–19–22–18–24–16–14–30–22–25–27–29. Can he assume the mean
of aphids on each boll at 20 per day at a significance level of 5%?
3. In the previous exercise and for a t-test, are 20 aphids enough for this researcher’s
sample
size?
4. Based on a sample, the shoe size of some women in the cesarean ward and other
hospital
wards is according to Table
11.14.
A. Can the standard deviation of mother’s shoe size in the two groups be
considered
equal?
B. Can the mean size of mothers’ shoes be equal in both groups?
5. If in a sample of 20 female students 8 without eating breakfast and in a sample of
15
boys 5 without eating breakfast have come to the university, do the following
three methods and compare and interpret the results.
A. Find the confidence interval for the difference between the two proportions
of
“without breakfast”.
B. Test the equality of the proportion of students without breakfast among
bo
ys and girls.
C. By running the Chi-squared test, determine if not eating breakfast is related
to
gender
6. The number of births by months of the year is as follows:
January February March April May June July August September October November December
528 503 496 46 518 411 45 44 37 41 331 416
Assuming that the birth rate is the same in all months of the year, check the
observed frequencies with the theoretical distribution.
https://t.me/med1917
11.16 Exercises 271
7. Systolic blood pressure of 16 patients was measured with two manometers A
and B and is shown in Table
11.15. Can the difference between the two methods
be ignored at a significance level of 5%?
8. The mean and standard deviation of adult height in a population by gender are
estimated and are given in Table
11.16. Can height be equal in men and women?
9. According to Table 3.2, are albumin levels equal in two groups of women?
10. According to Table 3.3, are the levels of urea acid equal in two groups of women?
11. According to Table 2.11 and assuming that the first and second groups are
the same subjects whose calcium levels were measured before and after an
experiment, is it permissible to consider the difference in calcium levels before
and after this experiment as non-significant?
12. Do the observations of Table 2.10 confirm the claim that the average prescribed
painkillers for each version are 2 items?
13. One cigarette factory claims that its cigarettes contain a maximum of 40 mg of
nicotine. The mean and standard deviation of 5 random samples of the products
of this factory are 42.6 mg and 3.7 mg. If we assume that the nicotine content
of cigarettes has a normal distribution. Test the claim of this factory at the
significance level of 0.01.
Table 11.15 Systolic blood pressure of 16 patients with two manometers A and B
Patient A B Patient A B
1 220 190 9 190 174
2 221 202 10 182 195
3 210 180 11 162 160
4 216 208 12 150 168
5 204 188 13 160 172
6 194 182 14 158 150
7 191 175 15 188 204
8 182 168 16 183 200
Table 11.16 Mean and
standard deviation of adult
height (Cm) in a population
by gender
Gender Mean Sd Sample n
Women 171.5 5.72 81
Men 179.1 5.85 64
https://t.me/med1917
272 11 Hypothesis Testing
Table 11.17 Daily protein intake in grams over a 24-hour period for two income groups: below
the poverty line (Group A) and above the poverty line (Group B)
Group A: 87.7-69.7-56.4-77.2-98.6-78.1-68.6-80.2-59.7-69.0-73.6
Group B: 73.7-55.0-65.8-76.7-72.0-49.7-51.4-86.0-73.3-62.0-65.4-65.5-75.8-62.1
14. A study was conducted to evaluate the effect of diet with exercise (group A)
versus diet without exercise (group B) on obese people. In this study, 37 people
from group A and 36 people from group B were selected and the results of
weight loss in these two groups are given below:
Group Mean Sd Sample n
A 7.63 1.59 37
B 7.79 2.36 36
Compare the effect between these two interventions at the significance level
of 0.05.
15. A study was carried out to compare daily protein uptake with income level. In
this study, the individuals were divided into two groups below the poverty line
(group A) and above the poverty line (group B). The protein uptake in grams
in a 24-h period is shown in Table
11.17:
A. Exam the equality of the variances of the two groups.
B. Are the mean of protein uptake equal in two groups at 5% significant level?
https://t.me/med1917
Chapter 12
Correlation and Regression
12.1 Introduction
In previous chapters, we’ve covered two main areas of inferential statistics:
confidence intervals and hypothesis tests. Another important aspect of inferential
statistics involves examining whether there’s a relationship between two or more
variables. For instance, we might want to know if there’s a relationship between a
person’s age and their blood pressure, or if the number of hours a student studies is
linked to their exam scores. When we collect observations on two variables for a
group of individuals (such as their height and weight), we turn to bivariate analysis
to explore how one variable is influenced by another.
Correlation and regression are two fundamental techniques used to assess the
strength of association between two variables. Correlation analysis helps determine
if there’s a linear relationship between variables, offering a unit-less measure of
association, typically on a linear scale. On the other hand, regression analysis delves
deeper into describing the nature of the relationship between variables, whether
it’s positive or negative, linear or nonlinear. Regression analysis also provides a
framework for predicting one variable (the dependent variable) based on another
(the predictor variable).
For these methods, one of the following goals can be enumerated:
•
Initial knowledge and understanding of the scope of the subject.
•
Finding the reasons and justifications for the phenomenon studied.
•
Trying to predict the future status of the studied variable.
In this chapter, we will explain about the relationship between two variables.
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024
S. H. Saneii and H. Doosti, Practical Biostatistics for Medical and Health Sciences,
https://doi.org/10.1007/978-981-97-3083-4_12
273
https://t.me/med1917
274 12 Correlation and Regression
Tip
Correlation and causality are two different concepts. Correlation is the
mutual relationship between two things, while causality is the reason for
causing something.
12.2 Correlation
Correlation, as discussed earlier, describes the statistical relationship between two
variables, indicating how they change together. It’s important to note that correlation
and causality are distinct concepts. While correlation reveals a mutual relationship
between variables, causality delves into the reasons behind causing something.
Tip
The closer the correlation coefficient is to 1, the more perfect and stronger it is.
On the contrary, the closer it is to zero, the weaker it is and the more indication
of absence of linear correlation.
Correlations possess two key properties: strength and direction. The strength of
a correlation is quantified by the correlation coefficient, which ranges between 0
and 1. A correlation coefficient of 1 signifies perfect correlation, while 0 denotes no
linear correlation at all. The direction of the correlation is determined by whether
it’s positive or negative.
Positive correlation occurs when both variables move in the same direction. This
means that as one variable increases, the other also increases, and vice versa.
Graphically, positive correlation is depicted by points rising from left to right on a
scatter plot.
Conversely, negative correlation arises when the variables move in opposite
directions. Here, as one variable increases, the other decreases, and vice versa.
Visually, negative correlation is represented by points falling while moving from
left to right on a scatter plot.
Tip
There is no rule for determining what size of correlation is considered strong,
moderate, or weak.
https://t.me/med1917
12.2 Correlation 275
No Correlation. When data points are dispersed throughout a scatter graph without
forming any clear pattern, it signifies the absence of correlation between the variables.
In this scenario, it becomes difficult to discern any consistent trend or directionality
in the relationship between the variables. The randomness in the distribution of points
indicates that there is no meaningful relationship or association between the variables
being studied.
Regardless of being the positive and negative correlations, the strength of
correlation can be defined as follows in terms of perfect correlation, strong
correlation, moderate correlation, and weak correlation. There is no rule for
determining what size of correlation is considered strong, moderate, or weak.
Therefore, this classification is based on the researcher’s personal experiences and
others may use other categories.
Perfect correlation shows convergence to complete and matched variations of two
variables together. So that, both variables change with the same exact proportion. The
value of perfect correlation is |1.0| and indicates that every change in the x variable
is accompanied by a proportional and corresponding change in the y variable and
vice versa. Perfect correlation could be negative (in different direction) or positive
(the same direction). Some examples of perfect positive linear correlation are: ( 1)
relation between Fahrenheit temperature and Celsius temperature. (2) The number
of patients who go to visit a clinician and the total income of clinician, (3) number
of kilometers traveled and gasoline consumption for your car. (4) Number of Covid-
19 patients and the number of deaths from Covid-19, because number of death is a
function of mortality rate. When plotted on a graph, correlation coefficient equals
to 1. Some examples of perfect negative linear correlation are: (1) The number of
daily consumed packs of cigarettes and people’s life expectancy has perfect negative
correlation. (2) The slower you drive (speed) in your car, the longer it will take you
(time) to reach your destination. (3) Two variables that have perfect negative linear
correlation are the distance from a door and the height of a wheelchair ramp. Because
every inch of ramp height requires 12 inches of distance to the door.
Packs of cigarette Life expectancy
0 85
0.5 81
1.0 77
1.5 73
2.0 69
2.5 65
https://t.me/med1917
276 12 Correlation and Regression
Strong correlation shows convergence to similar changes between the two variables.
Both variables change with the same fashion and trend. Strong correlation value
varies between |0.70| and larger thought to perfect correlation and shows that when
the variable X changes one unit, the Y variable changes not equally, but by a similar
value, and vice versa. The same value may be less than one unit or more than one
(for example, greater than 0.70 and less than 1.30 units).
Moderate correlation between the two variables is somewhat diluted for “aligned”
changes with each other, so that the two variables change in the same direction but
not much. The moderate correlation value between |35.0| 0.70 and larger thought to
strong correlation and shows that when the variable X changes with one unit, the
variable Y varies by almost half.
Weak correlation is a dilute convergence and less clear for changes between two
variables, so that the two variables change but slightly and minimal. Weak correlation
value is less than |0.35| and shows that when the variable X changes with one unit,
the Y variable changes by a smaller amount of variation (Fig.
12.1).
The scatter diagram is a very good visual tool for having a general glance of how
the relationship between the two variables is.
Correlation and correlation coefficient indicate the relationship between the two
variables and its numerical value, which varies between −1 and + 1. Correlation
coefficient is a measure used to express similarities between two distributions (two
sets of data). The correlation coefficient in the population is represented by the ρ (Rho)
and its sample estimation with r. The sign (+) or (−) in the correlation coefficient
expresses the direction of changes.
Positive correlation coefficient indicates direct correlation between the two
variables. That is, the increase and decrease of both variables are in the same
direction. If one variable increases, the other variable increases. If one of the
variables decreases, the other variable decreases.
Negative correlation coefficient indicates the inverse correlation between the two
variables. That is, two variables increase or decrease in the opposite direction. When
one of the variables increases, the other variable decreases. If one of the variables
decreases, the other variable increases.
https://t.me/med1917
12.2 Correlation 277
Fig. 12.1 Different patterns of correlation between two variables with scatter gram
When the correlation coefficient is zero or close to zero, it means that the
changes of one variable do not provide any information about the changes of the
other variable. The correlation coefficient of zero should not be considered as
independency of two variables. Because correlation only examines the linear
relationship between the two variables and does not address the nonlinear
relationship of variables. However, the correlation coefficient of two independent
variables is always zero. But the conclusions on the contrary are not true.
12.2.1 Pearson Correlation Coefficient
The most common coefficient of correlation is Pearson’s correlation coefficient and
is used for observations on a quantitative scale (interval, ratio). Pearson correlation
coefficient for variables X and Y is obtained using the following formula:
ρ =
Cov
(
x, y
)
√
var
(
x
)
.var
(
y
)
=
σ
xy
σ
x
.σ
y
=
N
i=1
(
x
i
− μ
x
)
y
i
− μ
y
N
i=1
(x
i
− μ
x
)
2
N
i=1
(y
i
− μ
y
)
2
https://t.me/med1917
278 12 Correlation and Regression
The above formula looks somewhat complicated, but using the practical formula
below makes correlation coefficient in sample somewhat easier to determine the
value of r.
ˆρ = r
xy
=
1
n − 1
x −
x
S
x
y −
y
S
y
=
n
x.y −
x
y
n
x
2
−
x
2
n
y
2
−
y
2
This formula may be seen as follows:
r
xy
=
x
i
.y
i
− x.y
(
n − 1
)
.S
x
.S
y
Even to simplify calculations, it is recommended to make a table such as Table 12.2
in which five columns are assigned for x, y, and. Then replace the sum of each column
directly in the above formula.
Tip
The correlation coefficient is very sensitive to outliers in the sense that a single
outlier could dramatically affect its value. It may be misleading when outliers
are present.
Assumptions for the Pearson Correlation Coefficient
Before applying the Pearson correlation to a dataset, it’s essential to examine the
following assumptions.
•
The data is selected through a random sampling process.
•
The relationship between the two datasets is linear and measured on an interval
or ratio scale.
•
The variables exhibit a joint normal distribution. This implies that given any
specific value of one variable, the values of the other variable are normally
distributed, and vice versa.
•
There are no outliers present in the data, which could significantly skew the results
of the correlation analysis.
Example 12.1 The education office aims to investigate the potential relationship
between the number of study hours and the final test scores on an exam. To do so,
they randomly select a s ample of 13 faculty students, as shown in Table
12.1.The
objective is to estimate the correlation coefficient between the study hours and the
corresponding final grades achieved in the exam.
Tip
Round the value of correlation coefficient r to three decimal places.
https://t.me/med1917
Соседние файлы в папке Библиотека им академика М.И. Перельмана