Biostatistics course Part 3 Data, summary and presentation

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Biostatistics coursePart 3Data, summary and
presentation

• Dr. en C. Nicolás Padilla Raygoza
• Facultad de Enfermería y Obstetricia de Celaya
• Universidad de Guanajuato México

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Biosketch

• Medical Doctor by University Autonomous of
  Guadalajara.
• Pediatrician by the Mexican Council of
  Certification on Pediatrics.
• Postgraduate Diploma on Epidemiology, London
  School of Hygine and Tropical Medicine,
  University of London.
• Master Sciences with aim in Epidemiology,
  Atlantic International University.
• Doctorate Sciences with aim in Epidemiology,
  Atlantic International University.
• Associated Professor B, School of Nursing and
  Obstetrics of Celaya, university of Guanajuato.
• padillawarm_at_gmail.com

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Competencies

• The reader will describe type of variables.
• He (she) will analyze how summary shows the
  different variables
• He (she) will calculate central trend measures
  and find them in graphics.
• He (she) will calculate dispersion measures and
  find them in graphics.

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Definitions

• Data are collected on the specific
  characteristics of each subject, and groups are
  formed to be compared.
• These characteristics are called variables,
  because they can change from each subject.
• Variable is obtained because it is
• A result of interest - dependent variable
• Or it explain the dependent variable - risk
  factor - independent variable.

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Type of data

• Classification for its measurement scale
• Qualititative
• Binary - dichotomous
• Ordinal
• Nominal
• Quantitative
• Discrete
• Continuous

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Type of data - Examples

• Qualitative
• Dichotomous - binary
• Gender male or female.
• Employment status employment or without
  employment.
• Ordinal
• Socioeconomic level high, medium, low.
• Nominal
• Residency place center, North, South, East,
  West.
• Civil status single, married, widowed, divorced,
  free union.
• Quantitative
• Discrete
• Number of offspring 1,2,3,4.
• Continuous
• Glucose in blood level 110 mg/dl, 145 mg/dl.

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Data summary

• Generally, we want to show the data in a summary
  form.
• Number of times that an event occur, is of our
  interest, it show us the variable distribution.
• We can generate a frequency list quantitative or
  qualitative.

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Summary of categorical data

• We can obtain frequencies of categorical data and
  summary them in a table or graphic.
• Example we have 21 agents of parasitic diseases
  isolated from children.

Giardia lamblia Entamoeba histolytica Ascaris
lumbricoides Enterobius vermicularis Ascaris
lumbricoides Enterobius vermicularis Giardia
lamblia
Giardia lamblia Entamoeba histolytica Ascaris
lumbricoides Enterobius vermicularis Ascaris
lumbricoides Enterobius vermicularis Giardia
lamblia
Giardia lamblia Entamoeba histolytica Ascaris
lumbricoides Enterobius vermicularis Ascaris
lumbricoides Enterobius vermicularis Giardia
lamblia
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Summary of categorical data

• List of parasites detected show us an idea of the
  frequency of each parasite, but that is not
  clear.
• If we ordered them, the idea is more clear.

Giardia lamblia Giardia lamblia Giardia
lamblia Giardia lamblia Giardia lamblia Giardia
lamblia Ascaris lumbricoides
Ascaris lumbricoides Ascaris lumbricoides Ascaris
lumbricoides Ascaris lumbricoides Ascaris
lumbricoides Enterobius vermicularis Enterobius
vermicularis
Enterobius vermicularis Enterobius
vermicularis Enterobius vermicularis Enterobius
vermicularis Entamoeba histolytica Entamoeba
histolytica Entamoeba histolytica
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Summary of categorical data

• We can show the results in a frequency
  distribution.

Frequency distribution of intestinal parasites
detected in children from CAISES Celaya, n21
Source Laboratory report
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Summary of categorical data

• It is useful to show the frequency of each
  category, expressed as percentage of the total
  frequency.
• It is called distribution of relative frequencies.

Frequency distribution of intestinal parasites
detected in children from CAISES Celaya, n21
Source Laboratory report
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Summary of categorical data

• Sometimes, the number of categories is high and
  should diminish the number of categories.

Distribution by death cause in Celaya, Gto,
during 2007
Source Certification of deaths
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Frequency distributions for quantitative data

• With quantitative data, we need group the data,
  before of show it in a frequencies or relative
  frequencies table.

Distribution of frequencies in students of FEOC
that have smoked at least once. n534
Source Health survey
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Frequency distributions for quantitative data

• With quantitative data, it is useful calculate
  cumulative frequency.

Distribution of frequencies in students of FEOC
that have smoked at least once. n534
Source Health survey
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Distributions of frequencies for grouped
quantitative data.

• Frequently, there are many categories with
  quantitative data, and we have to calculate
  intervals for each category.

Distribution of frequencies of ages of children
with acute streptoccocal pharyngotonsillitis
Source Padilla N, Moreno M. Comparison between
clarithromycin, azithromycin and propicillin in
the management of acute streptococcal
pharyngotonsillitis in children. Archivos de
Investigación Pediátrica de México 2005 85-11.
(In Spanish)
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Distribución de frecuencias para datos
cuantitativos agrupados
Distribution of frequencies of ages of children
with acute streptoccocal pharyngotonsillitis
Source Padilla N, Moreno M. Comparison between
clarithromycin, azithromycin and propicillin in
the management of acute streptococcal
pharyngotonsillitis in children. Archivos de
Investigación Pediátrica de México 2005 85-11.
(In Spanish)
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To group data

• Guide
• To obtain minimum and maximum values and decide
  the number of intervals.
• Number of intervals between 5 and 15.
• To assure interval limits.
• To assure that width of intervals been the same.
• To avoid that first or last interval been open.

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Charts

• Categorical data
• Bar chart
• Gráfica de pastel
• Quantitative data
• Histogram
• Polygon of frequencies

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Bar chart

• The frequency or relative frequency of a
  categorical variable can be show easily in a bar
  chart.
• It is used with categorical or numerical discrete
  data.
• Each bar represent one category and its high is
  the frequency or relative frequency.
• Bars should be separated.
• It is very important that Y axis begin with 0.

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Bar chart
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Grouped bar chart

• If we have a nominal categorical variable,
  divided in two categories, can show data with a
  grouped bar chart.
• It allow easy comparison between groups.

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Grouped bar chart
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Pie chart

• It is an alternative to show categorical
  variable.
• Each slice of pie correspond at frequency or
  relative frequency of categories of variable.
• It only shows one variable in each pie chart.
• If we want to make comparisons, we need to build
  two pie charts.

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Pie chart
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Pie chart
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Distribution of frequency charts histograms

• It is useful to quantitative variables.
• There are not spaces between bars.
• The area bar, not its high, represent its
  frequency.
• X axis should be continuous.
• Y axis should begin in 0.
• Width represent the interval for each group.

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Distribution of frequency charts histograms
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Distribution of frequency charts frequencies
polygon

• It is another form to show the frequency
  distribution of a numerical variable.
• It is building, joining the middle point higher
  of each bar of histogram.
• We should be take into account the width of each
  bar.
• We can plot more than one polygon in each chart,
  to make comparisons.

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Distribution of frequency charts polygon of
frequencies
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Distribution of frequencies cumulative histogram

• We can plot directly from a cumulative
  frequencies table.
• It is not necessary to make adjustments to the
  high of the bars, because the cumulative
  frequencies represent the total frequency
  superior, including the superior limit of the
  interval.

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Distribution of frequencies cumulative histogram
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Distribution of frequencies cumulative polygon
of frequencies

• We use them to see proportions below o above of a
  point in the curve.
• We can read median and percentiles, directly.
• If the distribution is symmetrical, it has S form
  symmetrical.
• If it is skewed to the right or to the left, will
  be flatten in that side.

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Distribution of frequencies cumulative polygon
of frequencies
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Other charts tree and leafs

• We use it to show directly quantitative data or
  preliminary step in the build a frequency
  distribution.
• We organize data determining the number of
  divisions (5-15).
• We plot a vertical line and put the first digit
  of category to the left of the line (tree) and
  the second digit to the right of the vertical
  line (leafs).

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Other charts tree and leafs
3 5 2 4 932 5 487 6 14
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Other charts box and line

• We plot a vertical line that represents the range
  of distribution.
• We plot a horizontal line that represents third
  quartile and another that represents the first
  quartile (box).
• The point middle of distribution is show as a
  horizontal line in the center of box.

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Other charts box and line
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Localization measures

• For categorical variable percentage
• For quantitative variable
• Central trend measures
• Mean
• Median
• Mode
• Dispersion measures
• Standard deviation
• Percentiles
• Range

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Central trend measures

• Mean
• It is the conventional mean.
• If we say that n observations have a xi value,
  then the value of the mean will be

_ X Sxi/n
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Central trend measures in a frequency distribution

• Each value of data (xi) occur with a frequency
  (fi), then
• Ina grouped distribution, we use point middle of
  each interval as x value.

_ X Sxifi/n
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Central trend measures in a frequency distribution
Interval Point middle Frequency
(fi) _________________________________ 1 3
2 18 4 6
5 27 7 9
8 34 10 12
11 22 13 15
14 13 ____________________
_____________ Total
114 Example of mean for a grouped
distribution (2 x 18) (5 x 27)
(8 x 34) (11 x 22) (14 x 13) 36 135
272 242 182 867 Mean
--------------------------------------------------
------------------- ----------------------------
------------ -------- 7.61
(18 27 34 22 13)
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114 Mean 7.61 years
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Central trend measures

• Median
• It is the value that divide the distribution in
  two equal parts.
• If it is a pair number of observations, the
  central values are summed and divided by two.

51.2, 53.5, 55.6, 65.0, 74.2 median is the value
at the half, thus Median 55.6 51.2, 53.5,
55.6, 61.4, 65.0, 74.2, 55.6 61.4 /2 Median
58.5
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Central trend measures for frequency distributions

• Median
• It is the value where is 50.

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Central trend measures

• Mode
• It is the value that occur more frequently.

Interval Point middle Frequency
(fi) _________________________________ 1 3
2 18 4 6
5 27 7 9
8 34 10 12
11 22 13 15
14 13 ____________________
_____________ Total
114
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Central trend measures

• Properties
• Mean is sensitive to the tails, median and mode,
  not.
• Mode can be affected by little changes in the
  data, median and mean, not.
• Mode and median can be find in a chart.
• The three measures are the same in a Normal
  distribution.

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Central trend measures

• What measurement to use?
• For skewed distributions, we use median.
• For statistical analysis or inference, we use
  mean.

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Dispersion measures

• Range
• It show the minimum and maximum values and the
  difference between they.

51.2, 53.5, 55.6, 61.4, 65.0, 74.2 Range of this
distribution es 51.2 74.2 kg. However, the
extreme values of this distribution are far
center of distribution, it unclear the fact that
the most data are between 53.5 and 65 kg.
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Dispersion measures

• Percentiles
• A percentile o centile is the value, below of
  which, a percentage given of data, has occurred.

See the distribution of stature in this
population. What is the range, median, percentile
25 and 75? Stature (cm.).
n Relative frequency ()
Cumulative frequency () 151
2
0.7
0.7 152
3 1.1
1.8 152
6
2.2
4.0 154
12 4.5
8.5 155
27
10.0
18.5 157
29 10.8
29.3 158
26
9.7
39.0 159
33 12.3
51.3 163
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13.8
65.1 164
16 5.9
71.0 165
24
8.9
79.9 168
18 6.7
86.6 169
14
5.2
91.8 171
6 2.2
94.0 174
7
2.6
96.6 175
1 0.4
97.0 177
4
1.5
98.5 179
2 0.7
99.2 184
1
0.4
99.6 185
1 0.4
100.0 _____________________
________________________________________________ T
otal 269
100.0
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Dispersion measures

• Standard deviation
• It is the more common form of to quantify the
  variability of a distribution.
• It measure the distance between each valu and its
  mean.

Subject High Value

S Xi - X 1 1.6 -1
Mean deviation -------------
2 1.7 0
n
3 1.8 1

_
X 1.7 Mean deviation
(-1)(0)(1)/3 0
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Dispersion measures

• Standard deviation
• We should be interest in magnitude of
  observations.
• If squared each deviation, we shall have positive
  values.
• If divided this add by n -1, we shall obtain
  variance and if we obtain square root, shall have
  standard deviation.

Subject High
Value2
S (Xi - X)2 1 1.6
0.1 Standard deviation v
--------------- 2 1.7
0
n-1 3
1.8 0.1
_
X 1.7
Standard deviation v0.2/2 0.32
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Dispersion measures fo grouped data

• Standard deviation
• It use the mean point of each interval.

S f(Xi - X)2
Standard deviation v
--------------
f - 1
Also, it can be
expressed as
Sfx2 - (Sfx)2 /Sf
Standard deviation v -------------------
--
S f -1

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Dispersion measures for grouped data

• For data with Normal distribution
• Around 68 of data are between -1 and 1 standard
  deviation.
• Around 95 of data are between -2 and 2 standard
  deviations.
• Around 99.9 of data are between -3 and 3
  standard deviations.
• Standard deviation is a measure of the width of
  the distribution. If the standard deviation
  change, the distribution change, also.

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Bibliography

• 1.- Kirkwood BR. Essentials of medical
  statistics. Oxford, Blackwell Science, 1988.
• 2.- Altman DG. Practical statistics for medical
  research. Boca Ratón, Chapman Hall/ CRC 1991.