Download - Introduction to Statistics 1 COD
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Overview Descriptive Statistics & Graphical Presentation of
Data Statistical Inference
Hypothesis Tests & Confidence Intervals T-tests (Paired/Two-sample) Regression (SLR & Multiple Regression) ANOVA/ANCOVA
Intended as an interview. Will provide slides after lectures
What’s in the lectures?...
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Lecture 1 Lecture 2 Lecture 3 Lecture 4 Descriptive Statistics and Graphical Presentation of Data1. Terminology
2. Frequency Distributions/Histograms3. Measures of data location 4. Measures of data spread5. Box-plots6. Scatter-plots 7. Clustering (Multivariate Data)
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Lecture 1 Lecture 2 Lecture 3 Lecture 4 Statistical Inference
1. Distributions & Densities2. Normal Distribution3. Sampling Distribution & Central Limit Theorem4. Hypothesis Tests5. P-values6. Confidence Intervals7. Two-Sample Inferences8. Paired Data
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Lecture 1 Lecture 2 Lecture 3 Lecture 4 Sample Inferences
1. Two-Sample Inferences Paired t-test Two-sample t-test
2. Inferences for more than two samples One-way ANOVA Two-way ANOVA Interactions in Two-way ANOVA
3. DataDesk demo
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Lecture 1 Lecture 2 Lecture 3 Lecture 4
1. Regression2. Correlation3. Multiple Regression4. ANCOVA5. Normality Checks6. Non-parametrics7. Sample Size Calculations8. Useful tools and websites
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Explanations of outputs
Videos with commentary
Help with deciding what test to use with what data
FIRST, A REALLY USEFUL SITE
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1. TerminologyPopulations & Samples
Population: the complete set of individuals, objects or scores of interest. Often too large to sample in its entirety It may be real or hypothetical (e.g. the results from an
experiment repeated ad infinitum)
Sample: A subset of the population. A sample may be classified as random (each member
has equal chance of being selected from a population) or convenience (what’s available).
Random selection attempts to ensure the sample is representative of the population.
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Variables Variables are the quantities measured in a
sample.They may be classified as: Quantitative i.e. numerical
Continuous (e.g. pH of a sample, patient cholesterol levels)
Discrete (e.g. number of bacteria colonies in a culture)
Categorical Nominal (e.g. gender, blood group) Ordinal (ranked e.g. mild, moderate or severe
illness). Often ordinal variables are re-coded to be quantitative.
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Variables
Variables can be further classified as: Dependent/Response. Variable of primary interest
(e.g. blood pressure in an antihypertensive drug trial). Not controlled by the experimenter.
Independent/Predictor called a Factor when controlled by experimenter. It
is often nominal (e.g. treatment) Covariate when not controlled.
If the value of a variable cannot be predicted in advance then the variable is referred to as a random variable
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Parameters & Statistics Parameters: Quantities that describe a
population characteristic. They are usually unknown and we wish to make statistical inferences about parameters. Different to perimeters.
Descriptive Statistics: Quantities and techniques used to describe a sample characteristic or illustrate the sample data e.g. mean, standard deviation, box-plot
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2. Frequency Distributions
An (Empirical) Frequency Distribution or Histogram for a continuous variable presents the counts of observations grouped within pre-specified classes or groups
A Relative Frequency Distribution presents the corresponding proportions of observations within the classes
A Barchart presents the frequencies for a categorical variable
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Example – Serum CK
Blood samples taken from 36 male volunteers as part of a study to determine the natural variation in CK concentration.
The serum CK concentrations were measured in (U/I) are as follows:
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Serum CK Data for 36 male volunteers
121 82 100 151 68 5895 145 64 201 101 16384 57 139 60 78 94
119 104 110 113 118 20362 83 67 93 92 11025 123 70 48 95 42
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Relative Frequency TableSerum CK
(U/I)Frequency Relative
FrequencyCumulative Rel.
Frequency20-39 1 0.028 0.02840-59 4 0.111 0.13960-79 7 0.194 0.33380-99 8 0.222 0.555100-119 8 0.222 0.777120-139 3 0.083 0.860140-159 2 0.056 0.916160-179 1 0.028 0.944180-199 0 0.000 0.944200-219 2 0.056 1.000Total 36 1.000
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Frequency Distribution
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4
6
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Freq
uenc
y
20 40 60 80 100 120 140 160 180 200 220
100.0%99.5%97.5%90.0%75.0%50.0%25.0%10.0%2.5%0.5%0.0%
maximum
quartilemedianquartile
minimum
203.00 203.00 203.00 154.60 118.75 94.50 67.25 54.30 25.00 25.00 25.00
QuantilesMeanStd DevStd Err Meanupper 95% Meanlower 95% MeanN
98.27777840.3807676.7301278111.9406684.614892
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Moments
CK-concentration-(U/l)
Distributions
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0.05
0.10
0.15
0.20
Rel
ativ
e Fr
eque
ncy
20 40 60 80 100 120 140 160 180 200 220
100.0%99.5%97.5%90.0%75.0%50.0%25.0%10.0%2.5%0.5%0.0%
maximum
quartilemedianquartile
minimum
203.00 203.00 203.00 154.60 118.75 94.50 67.25 54.30 25.00 25.00 25.00
QuantilesMeanStd DevStd Err Meanupper 95% Meanlower 95% MeanN
98.27777840.3807676.7301278111.9406684.614892
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Moments
CK-concentration-(U/l)
DistributionsRelative Frequency Distribution
Mode
Left tail
Right tail
(skewed)
Shaded area is percentage of males with CK values between 60 and 100 U/l, i.e. 42%.
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3. Measures of Central Tendency (Location)Measures of location indicate where on the number line the data are to be found. Common measures of location are:
(i) the Arithmetic Mean,(ii) the Median, and(iii) the Mode
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The Mean
Let x1,x2,x3,…,xn be the realised values of a random variable X, from a sample of size n. The sample arithmetic mean is defined as:
n
iin xx
1
1
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Example
Example 2: The systolic blood pressure of seven middle aged men were as follows:
151, 124, 132, 170, 146, 124 and 113.
The mean is
14.1377
113124146170132124151
x
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The Median and Mode
If the sample data are arranged in increasing order, the median is
(i) the middle value if n is an odd number, or(ii) midway between the two middle values if n is
an even number The mode is the most commonly occurring
value.
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Example 1 – n is odd
The reordered systolic blood pressure data seen earlier are:
113, 124, 124, 132, 146, 151, and 170.
The Median is the middle value of the ordered data, i.e. 132.
Two individuals have systolic blood pressure = 124 mm Hg, so the Mode is 124.
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Example 2 – n is even
Six men with high cholesterol participated in a study to investigate the effects of diet on cholesterol level. At the beginning of the study, their cholesterol levels (mg/dL) were as follows:
366, 327, 274, 292, 274 and 230.
Rearrange the data in numerical order as follows:
230, 274, 274, 292, 327 and 366.
The Median is half way between the middle two readings, i.e. (274+292) 2 = 283.
Two men have the same cholesterol level- the Mode is 274.
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Mean versus Median
Large sample values tend to inflate the mean. This will happen if the histogram of the data is right-skewed.
The median is not influenced by large sample values and is a better measure of centrality if the distribution is skewed.
Note if mean=median=mode then the data are said to be symmetrical
e.g. In the CK measurement study, the sample mean = 98.28. The median = 94.5, i.e. mean is larger than median indicating that mean is inflated by two large data values 201 and 203.
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4. Measures of Dispersion
Measures of dispersion characterise how spread out the distribution is, i.e., how variable the data are.
Commonly used measures of dispersion include:1. Range2. Variance & Standard deviation3. Coefficient of Variation (or relative standard
deviation)4. Inter-quartile range
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Range the sample Range is the difference
between the largest and smallest observations in the sample
easy to calculate; Blood pressure example: min=113 and
max=170, so the range=57 mmHg useful for “best” or “worst” case scenarios sensitive to extreme values
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Sample Variance
The sample variance, s2, is the arithmetic mean of the squared deviations from the sample mean:
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2
2
n
xxs
n
ii
>
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Standard Deviation
The sample standard deviation, s, is the square-root of the variance
s has the advantage of being in the same units as the original variable x
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2
n
xxs
n
ii
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Example
Data Deviation Deviation2
151 13.86 192.02124 -13.14 172.73132 -5.14 26.45170 32.86 1079.59146 8.86 78.45124 -13.14 172.73113 -24.14 582.88
Sum = 960.0 Sum = 0.00 Sum = 2304.8614.137x
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Example (contd.)
Therefore,
86.23047
1
2 i
i xx
6.191786.2304
s
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Coefficient of Variation The coefficient of variation (CV) or relative
standard deviation (RSD) is the sample standard deviation expressed as a percentage of the mean, i.e.
The CV is not affected by multiplicative changes in scale
Consequently, a useful way of comparing the dispersion of variables measured on different scales
%100
xsCV
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Example
The CV of the blood pressure data is:
i.e., the standard deviation is 14.3% as large as the mean.
%3.14
%1.137
6.19100
CV
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Inter-quartile range
The Median divides a distribution into two halves.
The first and third quartiles (denoted Q1 and Q3) are defined as follows: 25% of the data lie below Q1 (and 75% is above Q1),
25% of the data lie above Q3 (and 75% is below Q3)
The inter-quartile range (IQR) is the difference between the first and third quartiles, i.e. IQR = Q3- Q1
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Example
The ordered blood pressure data is:
113 124 124 132 146 151 170
Q1 Q3
Inter Quartile Range (IQR) is 151-124 = 27
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60% of slides complete!
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5. Box-plots
A box-plot is a visual description of the distribution based on Minimum Q1 Median Q3 Maximum
Useful for comparing large sets of data
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Example 1
The pulse rates of 12 individuals arranged in increasing order are:
62, 64, 68, 70, 70, 74, 74, 76, 76, 78, 78, 80
Q1=(68+70)2 = 69, Q3=(76+78)2 = 77
IQR = (77 – 69) = 8
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Example 1: Box-plot
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Example 2: Box-plots of intensities from 11 gene expression arrays
AG_04659_AS.cel AG_11745_AS.cel KB_5828_AS.cel KB_8840_AS.cel
810
1214
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Outliers
An outlier is an observation which does not appear to belong with the other data
Outliers can arise because of a measurement or recording error or because of equipment failure during an experiment, etc.
An outlier might be indicative of a sub-population, e.g. an abnormally low or high value in a medical test could indicate presence of an illness in the patient.
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Outlier Boxplot
Re-define the upper and lower limits of the boxplots (the whisker lines) as:Lower limit = Q1-1.5IQR, andUpper limit = Q3+1.5IQR
Note that the lines may not go as far as these limits
If a data point is < lower limit or > upper limit, the data point is considered to be an outlier.
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Example – CK data
outliers
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6. Scatter-plot
Displays the relationship between two continuous variables
Useful in the early stage of analysis when exploring data and determining is a linear regression analysis is appropriate
May show outliers in your data
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Example 1: Age versus Systolic Blood Pressure in a Clinical Trial
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Example 2: Up-regulation/Down-regulation of gene expression across an array (Control Cy5 versus Disease Cy3)
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Example of a Scatter-plot matrix (multiple pair-wise plots)
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Other graphical representations
Dot-Plots, Stem-and-leaf plots Not visually appealing
Pie-chart Visually appealing, but hard to compare two datasets. Best
for 3 to 7 categories. A total must be specified. Violin-plots
=boxplot+smooth density Nice visual of data shape
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Clustering is useful for visualising multivariate data and uncovering patterns, often reducing its complexity
Clustering is especially useful for high-dimensional data (p>>n): hundreds or perhaps thousands of variables
An obvious areas of application are gel electrophoresis and microarray experiments where the variables are protein abundances or gene expression ratios
Multivariate Data
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7. Clustering Aim: Find groups of samples or variables sharing
similiarity
Clustering requires a definition of distance between objects, quantifying a notion of (dis)similarity Points are grouped on the basis on minimum distance
apart (distance measures)
Once a pair are grouped, they are combined into a single point (using a linkage method) e.g. take their average. The process is then repeated.
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Clustering Clustering can be applied to rows or columns of a data set
(matrix) i.e. to the samples or variables
A tree can be constructed with branch length proportional to distances between linked clusters, called a Dendrogram
Clustering is an example of unsupervised learning: No use is made of sample annotations i.e. treatment groups, diagnosis groups
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UPGMA
Unweighted Pair-Group Method Average Most commonly used clustering method Procedure:
1. Each observation forms its own cluster 2. The two with minimum distance are grouped into a single
cluster representing a new observation- take their average 3. Repeat 2. until all data points form a single cluster
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Contrived Example
Array1 Array2 Array3
p53 9 3 7
mdm2 10 2 9
bcl2 1 9 4
cyclinE 6 5 5
caspase 8 1 10 3
5 genes of interest on 3 replicates arrays/gels
Calculate distance between each pair of genes
5.2)97()23()109()2,53(.. 222 mdmpdge
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Example
Construct a distance matrix of all pair-wise distances
p53 mdm2 bcl2 cyclinE caspase 8
p53 0 2.5 10.44 4.12 11.75
mdm2 - 0 12.5 6.4 13.93
bcl2 - - 0 6.48 1.41cyclinE - - - 0 7.35
caspase 8 - - - - 0
Cluster the 2 genes with smallest distance Take their average & re-calculate distances to other genes
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{p53 & mdm2} cyclin E {caspase-8 &
bcl-2}
{p53 & mdm2} 0 3.7 9.2
cyclin E 0 6.9
{caspase-8 & bcl-2} 0
p53 mdm2 cyclin E {caspase-8 & bcl-2}
p53 0 2.5 4.12 10.9
mdm2 0 6.4 9.1
cyclin E 0 6.9
{caspase-8 & bcl-2} 0
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Example (contd)
..and the final cluster:
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Example of a gene expression dendrogram
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Variety of approaches to clustering • Clustering techniques
– agglomerative -start with every element in its own cluster, and iteratively join clusters together
– divisive - start with one cluster and iteratively divide it into smaller clusters
• Distance Metrics– Euclidean (as-the-crow-flies)– Manhattan – Minkowski (a whole class of metrics)– Correlation (similarity in profiles: called similarity metrics)
• Linkage Rules – average: Use the mean distance between cluster members– single: Use the minimum distance (gives loose clusters)– complete: Use the maximum distance (gives tight clusters)– median: Use the median distance– centroid: Use the distance between the “average” member or
each cluster
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Clustering Summary
The clusters & tree topology often depend highly on the distance measure and linkage method used
Recommended to use two distance metrics, such as Euclidean and a correlation metric
A clustering algorithm will always yield clusters, whether the data are organised in clusters or not!