Download Descriptive statistics - Basic statistics for experimental researchers

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Faculty of Health Sciences
Descriptive statistics
Basic statistics for experimental researchers, Fall 2015
Julie Lyng Forman
Department of Biostatistics, University of Copenhagen
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Outline
Introduction
Describing categorical and discrete data
Describing continuous data
Tiny datasets
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Descriptive statistics
Metods for summarizing raw data
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Numerical descriptive statistics: Numbers.
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Graphical descriptive statistics: Figures.
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Folate with confidence limits
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Datatypes
In order to select an appropriate statistical method for analyzing
your data, you need to consider what type of data you have.
Quantitative variables:
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Continuous which (in principle) can take any value on a
continuous scale, e.g. concentrations.
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Discrete: which are most often counts (0,1,2,. . . ).
Categorical variables:
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Nominal, i.e. unordered grouping such as gender, genotype.
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Ordinal, i.e. ordered grouping such as tumor stage.
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Overview of descriptive methods
For categorical / discrete variables:
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Numerical: Tabulated values, percentages.
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Graphical: Barplots.
For continuous / discrete variables:
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Numerical: mean (average), standard deviation, quantiles.
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Graphical: Histogram, boxplot, stripchart.
Relation between two variables:
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Numerical: Cross tabulation, correlation.
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Graphical: Scatterplot, side by side plots.
Repeated measurements:
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Graphical: Spaghettiplots.
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Descriptive statistics with SAS
Compute summary statistics with
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PROC MEANS or PROC UNIVARIATE
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PROC FREQ for tablutation or cross-tabulation
A large range of plots can be made with PROC SGPLOT
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Scatterplots, histograms, boxplots, barplots, spaghettiplots
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PROC SGPANEL makes side by side plots over multiple
groups.
More about this at the SAS introduction later today . . .
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Outline
Introduction
Describing categorical and discrete data
Describing continuous data
Tiny datasets
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Summarizing categorical data
Example: Type 1 diabetes in NOD mice.
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Does treatment delay onset of diabetes?
Crosstabulated data:
treatment
diabetes (at 250 days follow-up)
Frequency|
Row Pct
|no
|yes
| Total
-----------+----------+--------+
control
|
3 |
8 |
11
| 27.27 | 72.73 |
-----------+----------+--------+
vehicle
|
7 |
7 |
14
| 50.00 | 50.00 |
-----------+--------+--------+
vorinostat |
15 |
2 |
17
| 88.24 | 11.76 |
---------+----------+--------+
Total
25
17
42
Reference Christensen et al, Lysine deacetylase inhibition prevents diabetes by
chromatin-independent immunoregulation and β-cell protection, PNAS,
www.pnas.org/cgi/doi/10.1073/pnas.1320850111
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Barplot for crosstabulated data
Used both for exploratory data analysis and for presentation.
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Side by side or stacked.
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Note: Total numbers, not percentages.
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Outline
Introduction
Describing categorical and discrete data
Describing continuous data
Tiny datasets
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Continuous data
Example: Hemoglobin in blood samples from 70 women (g/100ml)
10.2
12.9
13.7
12.7
11.7
11.6
13.3
13.3
10.5
11.4
11.2
10.2
10.8
9.7
10.6
12.9
14.6
8.8
14.7
13.1
11.0
12.1
13.5
11.1
11.3
11.6
12.3
12.2
9.3
12.9
10.9
13.0
13.4
11.8
12.0
12.1
12.5
12.9
13.1
11.0
13.4
11.4
10.7
10.9
12.0
11.7
11.9
15.1
13.5
11.5
10.8
13.6
11.2
11.1
14.9
13.2
10.3
11.9
14.6
10.4
9.4
14.1
11.4
10.4
13.7
12.1
11.8
10.6
10.9
12.5
We are interested in the distribution not the individual
measurements.
Reference: Kirkwood & Sterne, essential medical statistics, Blackwell
publishing, 2003.
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Histogram
Used when exploring data and for model checking.
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The volume of the box matches the proportion of data within
the interval. Do we see a normal distribution?
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Cumulative distribution
Main usage: Illustration in a basic statistics book.
Each step represents an observation at this value.
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Stepsize for hemoglobin is 1/70 × 100% = 1.43%.
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Cumulative distribution and quantiles
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Median: 11.85. Quartiles: 10.9 and 13.1.
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QQ-plots
Used for model checking
Empirical quantiles vs quantiles of a theoretical distribution.
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Hemoglobin data agree well with the normal distribution.
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Box and whiskers plot
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The box shows the median and the quartiles.
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The whiskers are drawn at minimum and maximum,
but not exceeding 1.5 times the length of the box.
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Data points beyond are highlighted as potential outliers.
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Overview of numerical methods
The most common summary statistics.
Central tendency: What is the most "typical" measurement?
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Median (50% above, 50% below)
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Average (aka the sample mean).
Variability: What is the typical distance to the center?
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Range (minimum – maximum).
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Quartiles (interval containing the central 50% of the data).
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Standard deviation (SD).
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Mean and standard deviation
The sample mean (or average) is:
P
x̄ =
n
x
=
x1 + x2 + x3 + · · · + xn
n
The standard deviation is:
sP
s=
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(x − x̄)2
(n − 1)
Note: division by degrees of freedom n − 1 rather than n.
Small values indicate that data is gathered closely around the
sample mean, while larger values indicate a larger variation
between subjects.
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Interpretation of the standard deviation
The normal range x̄ ± 2s contains ≈ 95% of the data in a normal
distribution. WARNING: not useful if data is skewed (next slide).
Example: For hemoglobin 11.98 ± 2 × 1.42 ' (9.14; 14.82).
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About normal ranges
The formula x̄ ± 2s is only valid if data have been sampled from a
normal distribtuion.
If the distribution is not normal, the computed normal range may
be misleading. Especially if the distribution is markedly skewed.
Instead estimate the normal range from the quantiles of the data
(2.5% -quantile ; 97.5% -quantile)
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This demands a fairly large sample size.
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Which summary statistics should I report?
Two descriptions of hemoglobin data:
Mean (SD): 11.98 (1.42).
Median (quartiles): 11.85 (10.9;13.1).
Mean (SD) is most often reported, but
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sensitive to extreme observations, outliers.
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if the distribution is skewed the inferred normal range is
wrong.
In these cases median and quartiles should be reported instead.
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Outline
Introduction
Describing categorical and discrete data
Describing continuous data
Tiny datasets
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Describing tiny datasets
Summary statistics are not sensible when sample size is very small!
You get a reasonably short and very accurate description of the
data by showing the full data:
6.11, 6.54, 8.75
while using summary statistics gives a false impression of what the
distribution looks like.
Mean (SD): 7.13 (1.41) - but is the data normal??
Median (quartiles): 6.54 (6.32;7.64) - but what is 25% of 3??.
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Visualisation of tiny datasets
Stripchart:
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Folate with confidence limits
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N2O+O2,op
O2,24h
Note: Overlayed confidence intervals for group means.