Transcript Statistics Primer Xiayu (Stacy) Huang Bioinformatics Shared Resource Email: [email protected] Sanford | Burnham Medical Research Institute.

Statistics Primer
Xiayu (Stacy) Huang
Bioinformatics Shared Resource
Email: [email protected]
Sanford | Burnham Medical Research Institute
Outline
 Overview of basic statistics
 Introduction
 Descriptive statistics
 Inferential statistics
 Most common statistical test and its applications
 T test
 Power analysis using t test
What is statistics?
 On American Statistical Association (ASA) website, statistics is
defined as the science of collection, analysis, interpretation and
presentation of data
 Using Statistics to make decision can be a double-edged sword
 In the 1980s, Marriott conducted an extensive survey with potential
customers on their attitudes about current hotel offerings. After
analyzing the data, the company launched Courtyard by Marriott,
which has been a huge success
 Coca-Cola performed a major consumer study in 1985 and, based on
the results, decided to reformulate Coke, its flagship drink. After a
huge public outcry, Coca-Cola had to backtrack and bring the original
formulation back to market
History of statistics
•17th-18th century
•Bernoulli number
•Bernoulli trial
•Bernoulli process
Jakob Bernoulli
•Bayes theorem
Thomas Bayes
•19th century
•Gaussian distribution
Carl Friedrich Gauss
•20th century
Karl Pearson
William Gosset
•Pearson correlation
•Chi-square distribution •Student’s t
Ronald Aylmer Fisher
•ANOVA, maximum likelihood
Why statistics is important to biologists?
• Designing experiment
How many ???
How many replicates for my microarray exp???
• Analyzing biological data and understanding analysis results
Identifying outlier
Normalization/transformation
Statistical test, etc.
DEGs
No replicates=No statistics?
• Preparing manuscript and grant applications
Study Scheme
Study Hypothesis
Design Study
Conduct Study and Collect data
Data Analysis
Summarizing data using
Descriptive Statistics
Choose Statistical
Test
Hypothesis Testing Using
Inferential Statistics
Compute test statistic
Compute p-value
Compare p-value and α
Make Conclusions
Branches of statistics
Descriptive statistics (Summary statistics)
 Summarize data graphically or numerically
 Lead to hypothesis generating
Inferential statistics
 Distinguish true difference from random variation
 Allow hypothesis testing
Types of data
Qualitative or
Quantitative
Example
Qualitative
Gender
Genotype
Tumor location
Qualitative or
Quantitative
Performance
Grade of tox
Disease stage
Quantitative
Age
Array intensities
Descriptive statistics—central tendency
 Mean—average
i.e. Age 24
27
22
25
24
23
28
23
25
26
22
29
24
25
26
27
28
29
Mean=(24+27+….+24)/13=24.8
 Median—middle value of sorted data
22
22
23
23
24
24
24
25
Median
 Mode—most frequently observed value
Mode is 24 with frequency of 3
Descriptive statistics—dispersion
 Range
i.e. Age
22
22
23
23
24
24
24
25
25
26
27
28
29
Range=highest value-lowest value=29-22=7
 Sample Variance (s2)\ Standard deviation (s)
(22  m ean )  (22  m ean )  ...  (29  m ean )
2
s 
2
s

2
13  1
2
 4.84
s  2.2
2
Values beyond two standard deviations from the mean can be considered as
“outliers” (>mean+2s=24.8+2x2.2=29.2 or <mean-2s=24.8-2x2.2=20.4)
 Standard error of mean (SEM)
SEM 
s
n

2.2
13
 0.61
Descriptive statistics—data distribution
 Histogram (x-bin, y-frequency)


Graphical representation showing the distribution of data
Summary graph showing how many data points falling in various ranges
22
22
23
23
Frequency table
Bin
Frequency
20-22
2
22-24
5
24-26
3
26-28
2
28-30
1
Percentage table
Bin
percentage
20-22
0.155
22-24
0.38
24-26
0.23
26-28
0.155
28-30
0.08
24
24
24
25
25
26
27
28
29
Histogram\frequency distribution
Histogram\probability distribution
Descriptive statistics—data distribution
 Different data distributions
Approximate normal distribution
i.e. height of people, length of dogs
Right skewed distribution
i.e. FC of Microarray data
Left skewed distribution
i.e. distribution of age at retirement
Normal (or Gaussian) distribution
mean=median=mode
•Bell-shaped curve
•Symmetrical about mean
•Mean, median and mode are equal
•~68% data points fall within 1 sd of mean
•~95% data points fall within 2 sd of mean
•~99.7% data points fall within 3 sd of mean
Installing graphpad prism
You can install Prism on Institute supplied computers, including home
and personal computers.
http://graphpad.com/paasl/index.cfm?sitecode=burnhm
SERIAL NUMBERS:
Macintosh version
contacting IT ([email protected]) to get serial number
Windows version
contacting IT ([email protected]) to get serial number
Calculating descriptive statistics in excel
Calculating descriptive statistics in prism
Calculating descriptive statistics in prism
Graphically displaying descriptive statistics
Histogram
Mean error bar plot
Line plot w/o error bar
Graphically displaying descriptive statistics
in Prism
Histogram and frequency distribution
Mean error bar plot
Graphically displaying descriptive statistics
in Prism
Group line plot
Group line plot without
error bar
Group line plot with
error bar
Choosing right measures of descriptive
statistics
Normal distribution
Skewed distribution
Normal distribution: mean and standard deviation
Skewed distribution: transform data to normal distribution
Outline
 Overview of basic statistics
 Brief Introduction
 Descriptive statistics
 Inferential statistics
 Most common statistical tests and its applications
 T test
 Power analysis using t test
Inferential statistics
Parametric
 Interval or ratio measurements
 Continuous variable
 Usually assuming data are normally distributed
Nonparametric
 Ordinal or nominal measurements
 Discreet variables
 Making no assumption about how data is distributed
Inferential statistics-hypothesis
Null hypothesis (H0)
new drug effect = old drug effect
tumor growth of MT = tumor growth of WT
Alternative hypothesis (HA)
•
•
is the opposite of null hypothesis
is generally the hypothesis that is believed to be
true by the researcher
new drug effect ≠ or > old drug effect
tumor growth of MT ≠ or < tumor growth of WT
Inferential statistics-one and two sided
tests
 Hypothesis tests can be one or two sided (tailed)
 One sided tests are directional:
H0 : new drug effect ≤ old drug effect
HA : new drug effect > old drug effect
 Two sided tests are not directional:
H0 : new drug effect = old drug effect
HA : new drug effect ≠ old drug effect
Inferential statistics-type I and type II
errors
“Actual situation”
No difference (H0)
No difference
“Measured”
Difference (HA)
Correct decision (TN)
1-α
Type II error (FN)
β
Type I error (FP)
α
Correct decision (TP)
1-β
Difference
FOB screening(bowel cancer)
“Actual situation”
“Measured”
-
+
-
1820
10
1830
+
180
20
200
2000
30
Correct decision (TN)
1-α=1820/2000=0.91
Type II error (FN)
FN=10/30=0.33
Type I error (FP)
α=180/2000=0.09
Correct decision (TP)
1-β=20/30=0.67
Inferential statistics-type I and type II
errors
• Control type I and type II errors
• Inverse relationship between type I and type II errors
• Make a choice to control which error
• i.e. controlling type I error (FP) is more important for
microarray data than type II error (FN)
• i.e. controlling type II error (FN) is more important for
cancer screening test than type I error (FP)
• Choose type I and type II errors for statistical test?
• Common choices (α = 5%, β = 20%)
• Exploratory study (α = 10%, β = 10%)
• Confirmatory study (α = 1%, β = 10%)
Inferential Statistics-P-value
• the probability that an observed difference could have occurred
by chance under null hypothesis
• Computed from test statistics score
• P-value is the same as false positive rate
• P-value below cut off (α) is referred as “statistically significant”
Inferential Statistics-Power
Power (1-β, aka true positive rate (TP))
• Probability of detecting a significant scientific difference
when it does exist
Power depends on:
 Sample size (n)
 Standard deviation (s)
 Size of the difference you want to detect (δ)
 False positive rate (α)

s
Effect size
Study scheme
Study Hypothesis
Design Study
Conduct Study and Collect data
Data Analysis
Calculating and Displaying
Descriptive Statistics
Choose Statistical Test
Hypothesis Testing Using
Inferential Statistics
Compute test statistic
Compute p-value
Compare p-value and α
Make Conclusions
How to choose an appropriate statistical test?
 Type of data


Quantitative
Qualitative
 Type of research question



Association
Correlation
Comparison
 Data structure



Independent
Paired
Matched
Statistical test decision making tree
For qualitative or nonnumerical data
For quantitative or numerical data
Statistical test decision making tree
Relationship
between
variables
Two sample comparison
Multiple
sample
comparison
Outline
 Overview of basic statistics
 Brief Introduction
 Descriptive statistics
 Inferential statistics
 Most common statistical test and its applications
 T test
 Power analysis using t test
Student’s t test
Guinness employee William Sealy Gosset
published the 'Student's t-test' in 1908
Types of t test
 One sample t test: test if a sample mean differs
significantly from the given known mean
 Unpaired t test: test if two independent sample means
differ significantly
 Paired t test: test if two dependent sample means
differ significantly (mean of pre and post treatment
for same set of patients
Application of t test in biology
Mincroarry experiment
WT
MT
Proteomics experiment
WT
MT
Biological reps
Technical reps
You need to have at least two replicates in each condition
to do t test, otherwise, t test is invalid and you won’t have statistics
Two sample unpaired t test
 Assumptions



Data is approximately normally distributed
The sample has been independently and randomly selected
Similar variances between comparing groups
 Hypothesis (two sided or one sided)
H 0 : 1   2  0
H A : 1   2  0
 Test statistics
( X 1  X 2 )  ( 1   2 )
t
sp
t  , n1  n 2  2
1 / n1  1 / n 2
( n 1  1) s1  ( n 2  1) s 2
2
sp 
2
n1  n 2  2
X 1, X 2
-- sample means
 1 ,  2 -- population means
s1 , s 2 -- sample standard deviation
2
n1 , n 2 -- sample size
sp
2
-- pooled sample variance
Sample data
1st Question to be answered:
Will the two treatments have different effect on patients’ remission time from cancer?
Patients
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Remission time
Treatments
from cancer
(years)
Drug
7
Drug
5
Drug
2
Drug
8
Drug
3
Drug
4
Drug
10
Drug
7
Drug
4
Drug
9
Placebo
4
Placebo
3
Placebo
1
Placebo
6
Placebo
2
Placebo
4
Placebo
9
Placebo
5
Placebo
3
Placebo
8
Summarizing sample data using
descriptive statistics
Hypothesis testing of sample data
using inferential statistics
Step1: Choosing an appropriate statistical test
Step2: Performing statistical test in software
Step3: Making conclusions
Statistical test decision making tree
Two sample t test in Prism-normality check
Two sample t test in Prism
Two sample t test in excel
Power analysis using two sample t test
2nd question to be answered:
How many patients do we need in order to detect a significantly difference
b/w two treatments?
s ( t1   / 2  t1   )
2
n 

2
2

( t1   / 2  t1   )
(

)
2
2
s
N





α
β
δ/s
Test
efficiency
K:1
imbalance
Power analysis of t test in G*power
Power analysis of t test in G*power
Basic Statistics tools
Statistics softwares and packages:
1.Excel and add-ins: EZAnalyze, Analysis Toolpak
2. Our institute supported Prism
3. SPSS, Statistica (commercial)
4. SAS (commercial) and R
5. G*Power
Basic statistics books:
1. Intro Stats, SDSU, 2nd edition, Deveaux, Velleman, Bock
2. Choosing and Using Statistics: A Biologist's Guide
3. Introduction to Statistics for Biology
4. Biostatistical analysis, fifth edition, Jerrold H. Zar
Statistics videos:
1. http://www.microbiologybytes.com/maths/videos
2. http://www.youtube.com: descriptive statistics, basic statistics,
install 2007 Excel data analysis add-ins…
Next.....
 My presentation will be posted on website:
http://bsrweb.burnham.org/
 I am located in building 10, Office 2405, ext 3916
 Feel free to come or call or send e-mail to ask
questions ([email protected])
 Group email: [email protected]