Data Analysis for Gene Chip Data Part I: Onegeneatatime methods

1

• Data Analysis for Gene Chip DataPart I
  One-gene-at-a-time methods
• Min-Te Chao
• 2002/10/28

2
Outline

• Simple description of gene chip data
• Earlier works
• Mutiple t-test and SAM
• Lees ANOVA
• Wongs factor models
• Efrons empirical Bayes

3
Remarks

• Most works are statistical analysis, not really
  machine learning type
• Very small set of training sample not to
  mention the test sample
• Medical research needs scientific rigor when we
  can

4
Arthritis and Rheumatism

• Guidelines for the submission and reviews of
  reports involving microarray technology
• v.46, no. 4, 859-861

5
Reproducibility

• Should document the accuracy and precision of
  data, including run-to-run variability of each
  gene
• No arbitrary setting of threshold (e.g., 2-fold)
• Careful evaluation of false discovery rate

6
Statistical Analysis

• Statistical analysis is absolutely necessary to
  support claims of an increase or decrease of gene
  expression
• Such rigor requires multiple experiments and
  analysis of standard statistical instruments.

7
Sample Heterogenenity

• Strongly recommends that investigators focus
  studies on homogenous cell populations until
  other methodological and data analysis problems
  can be resolved.

8
Independent Confirmation

• It is important that the findings be confirmed
  using an independent method, preferably with
  separate samples rather than restating of the
  original mRNA.

9
Microarray

• Other terms
• DNA array
• DNA chips
• biochips
• Gene chips

10

• The underlying principle is the same for all
  microarrays, no matter how they are made
• Gene function is the key element researchers want
  to extract from the sequence
• DNA array is one of the most important tools
• (Nature, v.416, April 2002 885-891)

11
2 types of microarray

• cDNA
• Oligonucleotides
• DIY type

12

• Microarray
• allows the researchers to determine which
  genes are being expressed in a given cell type at
  a particular time and under particular condition
• Gene-expression

13
Basic data form

• On each array, there are p spots (pgt1000,
  sometimes 20000). Each spot has k probes (k20 or
  so). There are usually 2k measurements
  (expressions) per spot, and the k differences, or
  the difference of logs, are used.
• Sometimes they only give you a summary
  statistics, e.g. median, mean,.. per spot

14

• Each spot corresponding to a gene
• For each study, we can arrange the chips so that
  the i-th spot represents the i-th gene. (genes
  close in index may not be close physically at
  all)
• This means that when we read the i-th spot of
  all chips in one study, we know we get different
  measurements of the same i-th gene

15

• Data of one chip can be arranged in a matrix
  form,
• Y X_1, X_2, , X_p
• Just as in a regression setup. But in practice, n
  (chips used) is small compared with p.
• Y is the response cell type, experimental
  condition, survival time,

16

• For a spot with 20 probes, see Efron et al.
  (2001, JASA, p.1153).

17
Earlier works

• Cluster analysis
• Fold methods
• Multiple t with Bonferroni correction

18
Multiple t with Bonferroni correction

• It is too conservative
• Family wise error rate
• Among G tests, the probability of at least one
  false reject basically goes to 1 with
  exponential rate in G

19

• Sidaks single-step adjusted p-value
• p1-(1-p)G
• Bonferronis single-step adjusted p-value
• pminGp,1
• All are very conservative

20
FDR false discovery rate

• Roughly Among all rejected cases, how many are
  rejected wrong?
• (Benjamini and Hochberg 1995 JRSSB, 289-300)
  Sequential p-method

21
Sequential p-method

• Using the observed data, it estimates the
  rejection regions so that the
• FDR lt alpha
• Order all p-values, from small to large, and
  obtain a k so the first k hypotheses (wrt the
  smallest k p-values) are rejected.

22

• Since we have a different definition for error to
  control, it will increase the power
• For modifications, see Storey (2002, JRSSB,
  479-498)
• These are criteria specifically designed to
  handle risk assessment when G is large

23
Role of permutation

• For tests (multiple or not), it is important to
  use a null distribution
• It is generated by a well-designed permutation
  (of the columns of the data matrix) column
  refers to observations, not genes.

24
One simple example

• Let us say we look at the first gene, with n_1
  arrays for treatment and n_2 arrays for control
• We use a t-statistics, t_1, say. What is the
  p-value corresponding to this observed t_1?

25

• Permute the nn_n_2 columns of data of the data
  matrix. Look at first row (corresponds to the
  first gene)
• Treat the first n_1 numbers as a fake
  treatment, the last n_2 numbers as a fake
  control , compute a t-value, say we get s_1

26

• Permute again and do the same thing and we get
  s_2, .
• Do it B times and get s_1, s_2, ., s_B
• Treat these ss as a (bootstrap) sample for the
  null distribution of the t_1 statistic
• The p-value of the earlier t_1 is found from the
  ecdf of the s_j, j1,2,,B

27

• Permutation plays a major role --- finding a
  reference measure of variation in various
  situations
• For a well designed experiment with microarray,
  DOE techniques will play an important role in
  determining how to do proper permutations.

28
SAM significance analysis of microarray

• A standard method of microarray analysis, taught
  many times in Stanford short courses of data
  mining
• Modified multiple t-tests
• Using the permutation of certain data columns to
  evaluate variation of data in each gene

29

• Original paper is hard to read
• (Tusher, Tibshirani and Chu, PNAS 2001, v.98,
  no.9, 5116-5121)
• But the SAM manual is a lot easier to read for
  statisticians (free software for academia use)

30

• D(i)X_treatment X_control over
• s(i)s_0
• i1,2,,G
• D(1)ltD(2)lt..
• Used in SAM, s_0 is a carefully determined
  constant gt0.

31

• D(i) are used with certain group of permutations
  of the columns D(i) are also ordered
• Plot D vs. D, points outside the 45-degree line
  by a threshold Delta are signals of significant
  expression change.
• Control the value of Delta to get different FDR.

32
Other model-based methods

• Wongs model
• PM-MM \theta \phi \epsilon
• Outlier detection
• Model validation
• Li and Wong (2001, PNAS v.98, no.1, 31-36)

33
Lees work

• ANOVA based
• May do unbalanced data e.g., 7 microarray chips
• (Lee et al. 2000, PNAS, v.97, 9834-9839)

34
Empirical Bayes

• (Efron et al. (2001) JASA, v.96, 1151-1160)
• Use a mix model
• f(z)p_0 f_0(z)p_1 f_1(z)
• with f_0, f_1 estimated by data.
• p_1prior prob that a gene expression is
  affected (by a treatment)

35

• A key idea is to use permuted (columns) data to
  estimate f_0
• Use a tricky logistic regression method
• Eventually found
• p_1(Z) the a posteriori probability that a
  gene at expression level Z is affected

36
Part I conclusion

• Earlier methods are relatively easy to
  understand, but to get familiar with the
  bio-language needs time
• More powerful data analytic methods will continue
  to develop
• It is important to first understand the basic
  problems of biologist before we jump with the
  fancy stat methods

37

• We may do the wrong problem
• But if the problem is relevant, even simple
  methods can get good recognition
• All methods so far are first moment only ie,
  not too much different from multiple t tests or,
  they all are one-gene-at-a-time methods.

38

• We did not address issues about data cleaning,
  outlier detection, normalization, etc. Microarray
  data are highly noisy, these problems are by no
  means trivial.
• As the cost per chip goes down, the number of
  chips per problem may grow. But still
  well-designed experiments, e.g., fractional
  factorial, has room to play in this game

39

• Statistical methods, as compared with machine
  learn based methods, will play a more important
  role for this type of data since, with a model,
  parametric or not, one can attach a measure of
  confidence to the claimed result. This is crucial
  for scientific development.

40
Quote

• The statistical literature for microarrays, still
  in its infancy and with much of it unpublished,
  has tended to focus on frequentist
  data-analytical devices, such as cluster
  analysis, bootstrapping and linear models.
  (Efron, B. 2001)