biostatistics course part 3 data, summary and presentation dr. en c. nicolas padilla raygoza...

Biostatistics coursePart 3

Data, summary and presentation

Dr. en C. Nicolas Padilla Raygoza

Department of Nursing and Obstetrics

Division of Health Sciences and Engineering

Campus Celaya Salvatierra

University of Guanajuato Mexico

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. Professor Titular A, Full Time, University of Guanajuato. Level 1 National Researcher System [email protected] [email protected]

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.

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.

Type of data

Classification for its measurement scale: Qualititative

Binary - dichotomous Ordinal Nominal

Quantitative Discrete Continuous

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.

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.

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



Giardia lamblia

Giardia lamblia






Giardia lamblia

Giardia lamblia






Giardia lamblia


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

















We can show the results in a frequency distribution.

Parasite n

Giardia lamblia 6

Ascaris lumbricoides 6

Enterobius vermicularis 6

Entamoeba histolytica 3

Total 21

Frequency distribution of intestinal parasites detected in children from CAISES Celaya, n=21

Source: Laboratory report

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.

Source: Laboratory report

Parásito n %

Giardia lamblia 6 28.57

Ascaris lumbricoides 6 28.57


6 28.57


3 14.29

Total 21 100.00

Frequency distribution of intestinal parasites detected in children from CAISES Celaya, n=21


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

Death cause n %

Cardiovascular disease 12,525 21.96

Cancer 10,321 18.10

Lower respiratory infections

8,745 15.34

Other 25,435 44.60

Total 57,026 100.00

Distribution by death cause in Celaya, Gto, during 2012

Source: Certification of deaths

Frequency distributions for quantitative data With quantitative data, we need group the data,

before of show it in a frequencies or relative frequencies table.

Age (years) n %

19 52 14.70

20 32 9.00

21 46 12.99

22 67 18.94

23 26 7.35

24 77 21.76

25 54 15.26

Total 534 100.00

Distribution of frequencies in students of FEOC that have smoked at least once. n=534

Source: Health survey

With quantitative data, it is useful calculate cumulative frequency.

Source: Health survey

Age (years) n % % cumulative

19 52 14.70 14.70

20 32 9.00 23.70

21 46 12.99 36.69

22 67 18.94 55.63

23 26 7.35 62.98

24 77 21.76 84.74

25 54 15.26 100.00

Total 534 100.00

Frequency distributions for quantitative data

Distribution of frequencies in students of Campus that have smoked at least once. n=534

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; 8:5-11. (In Spanish)

Age (years) n %

<1 2 0.51

1 8 2.00

2 13 3.30

3 29 7.36

4 37 9.39

5 44 11.17

6 51 12.94

7 50 12.69

8 49 12.44

9 32 8.12

10 25 6.35

11 22 5.58

12 14 3.55

13 9 2.28

14 7 1.78

15 2 0.51

Total 394 100.00

Distribución de frecuencias para datos cuantitativos agrupados

Age (years) n %

<1 - 3 52 13.20

4 - 6 132 33.50

6 - 9 131 33.25

10 - 12 61 15.48

13 - 15 18 4.57

Total 394 100.00

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; 8:5-11. (In Spanish)

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.

Categorical data Bar chart Gráfica de pastel

Quantitative data Histogram Polygon of frequencies

Charts

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.

Bar chart

Gastrintestinal infections

01234567

Cryptos. E.histolyt. E.coli Giardia Rotavirus Shigella

Agents

Freq

uenc

y

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.

Grouped bar chart

Gastrointestinal infections

0

1

2

3

4

5

Crypt. E.histolyt. E.coli Giardia Rotavirus Shigella

Agents

Freq

uenc

y Males

Females

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 or more pie charts.

Pie chart

Civil status of women in a community

Single28%

Married44%

Divorced11%

Widowed8%

Free union9%

Pie chart

Civil status of men in a community

Single31%

Married41%

Divorced

11%

Widowed

1%

Free union16%

Civil status of women in a community

Single28%

Married44%

Widowed

8%

Free union

9%

Divorced

11%

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.

Number of sons in women from Celaya

0

100

200

300

400

500

600

700

1 2 3 4 5 6 7 8+

Number of sons

Nu

mb

er

of

wo

ma

n

Distribution of frequency charts: histograms

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.

Number of sons of women from Celaya

0

100

200

300

400

500

600

700

1 2 3 4 5 6 7 8+

Number of sons

Nu

mb

er

of

wo

me

n

Distribution of frequency charts: polygon of frequencies

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.

Cumulative frequency of birthweight

0

20

40

60

80

100

120

501- 1001- 1501- 2001- 2501- 3001- 3501- 4001- 4501- 5000+

Weight

Cu

mu

lati

ve

freq

uen

cy (

%)

New borns

Distribution of frequencies: cumulative histogram

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.

Distribution of frequencies: cumulative polygon of frequencies

Cumulative frequencies of birthweight

0

20

40

60

80

100

120

501- 1001- 1501- 2001- 2501- 3001- 3501- 4001- 4501- 5000+

Weight

Cu

mu

lati

ve

freq

uen

cy (

%)

New borns

Distribution of frequencies: cumulative polygon of frequencies

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).

Other charts: tree and leafs

Patient

Age

1 54

2 35

3 49

4 61

5 58

6 64

7 32

8 57

9 43

10 42

3 5 2

4 932

5 487

6 14

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.

Other charts: box and line

5500

5000

4500

4000

3500

3000

2500

2000

1500

1000

500

Localization measures

For categorical variable: percentage For quantitative variable:

Central trend measures: Mean Median Mode

Dispersion measures: Standard deviation Percentiles Range

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 =Σxi/n

Each value of data (xi) occur with a frequency (fi), then:

In a grouped distribution, we use point middle of each interval as x value.

_X =Σxifi/n

Central trend measures in a frequency distribution

Central trend measures in a frequency distribution

Interval Point middle Frequency (fi)_________________________________

1 – 3 2 184 – 6 5 277 – 9 8 3410 – 12 11 2213 – 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 867Mean = --------------------------------------------------------------------- = ---------------------------------------- = -------- = 7.61 (18 + 27 + 34 + 22 + 13) 114 114

Mean = 7.61 years


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

Central trend measures for frequency distributions Median

It is the value where is 50%.

Cumulative frequency of birthweight

0

20

40

60

80

100

120

501- 1001- 1501- 2001- 2501- 3001- 3501- 4001- 4501- 5000+

Weight

Cu

mu

lati

ve

freq

uen

cy (

%)

New borns


Mode It is the value that occur more frequently.

Interval Point middle Frequency (fi)_________________________________

1 – 3 2 184 – 6 5 277 – 9 8 3410 – 12 11 2213 – 15 14 13

_________________________________Total 114


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.


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

mean.

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.

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 0.7152 3 1.1 1.8152 6 2.2 4.0154 12 4.5 8.5155 27 10.0 18.5157 29 10.8 29.3158 26 9.7 39.0159 33 12.3 51.3163 37 13.8 65.1164 16 5.9 71.0165 24 8.9 79.9168 18 6.7 86.6169 14 5.2 91.8171 6 2.2 94.0174 7 2.6 96.6175 1 0.4 97.0177 4 1.5 98.5179 2 0.7 99.2184 1 0.4 99.6185 1 0.4 100.0_____________________________________________________________________Total 269 100.0

Dispersion measures

Standard deviation It is the more common form of to quantify the

variability of a distribution. It measure the distance between each value

and its mean.

Subject High Value Σ 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

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

Σ (Xi - X)2 1 1.6 0.1 Standard deviation =√ --------------- 2 1.7 0 n-1 3 1.8 0.1 _ X= 1.7 Standard deviation = √0.2/2 = 0.32

Dispersion measures fo grouped data

Standard deviation It use the mean point of each interval.

Σ f(Xi - X)2 Standard deviation =√ -------------- f - 1 Also, it can be expressed as:

Σfx2 - (Σfx)2 /Σf Standard deviation = √ --------------------- Σ f -1

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.

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.

biostatistics course part 3 data, summary and presentation dr. en c. nicolas padilla raygoza...

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