Analysis of Differential Expression T-test ANOVA Non-parametric methods Correlation Regression.

Analysis of Differential Expression

T-testANOVANon-parametric methodsCorrelationRegression

Research Question

Do nicotine-exposed rats have different X gene expression than control rats in ventral tegmental area? Design an experiment in which treatment rats

(N>2) are exposed to nicotine and control rats (N>2) are exposed to saline.

Collect RNA from VTA, convert to cDNA Determine the amount of X transcript in each

individual. Perform a test of means considering the

variability within each group.

Observed difference between groups

May be due to Treatment Chance

Hypothesis Testing

Null hypothesis: There is no difference between the means of the groups.

Alternative hypothesis: Means of the groups are different.

Hypothesis testing

You can not accept null hypothesis You can reject it You can support it

P-value

The ‘P’ stands for probability, and measures how likely it is that any observed difference between groups is due to chance, alone.

P-value

there is a significant difference between groups if the P value is small enough (e.g., <0.05).

P value equals to the probability of type I error.

Type I error: wrongly concluding that there is a difference between groups (false positive).

Type II error: wrongly concluding that there is no difference between groups (false negative).

Multiple tests on the same data

Expression data on multiple genes from the same individuals

Subsets of genes are coregulated thus they are not independent.

Such data requires multiple tests.

Why not do multiple t-tests? Or if you do, adjust the p-values

Because it increases type I error: a study involving four treatments, there are

six possible pairwise comparisons. If the chance of a type I error in one such

comparison is 0.05, then the chance of not committing a type I error is 1 – 0.05 = 0.95.

then the chance of not committing a type I error in any one of them is 0.956 = 0.74.

Cumulative type I error = 1-0.74=0.26

Normal Distribution

it is entirely defined by two quantities: its mean and its standard deviation (SD). The mean determines where the peak

occurs and the SD determines the shape of the

curve.

Curves: same mean, different stds

Rules of normal distribution

68.3% of the distribution falls within 1 SD of the mean (i.e. between mean – SD and mean + SD);

95.4% of the distribution falls between mean – 2 SD and mean + 2 SD;

99.7% of the distribution falls between mean – 3 SD and mean + 3 SD.

Most commonly used rule

95% of the distribution falls between mean – 1.96 SD and mean + 1.96 SD

If the data are normally distributed, one can use a range (confidence interval) within which 95% of the data falls into.

A sample

Samples vary Samples are collected in limited

numbers They are representatives of a

population. A sample:

E.g., nicotine treated rat RNA

Sample means

Consider all possible samples of fixed size (n) drawn from a population.

Each of these samples has its own mean and these means will vary between samples.

Each sample will have their own distribution, thus their own std.

Population mean

The mean of all the sample means is equal to the population mean ().

SD of the sample means measures the deviation of individual sample means from the population mean ()

Standard error

It reflects the effect of sample size, larger the SE, either the variation is high or sample size is small.

Confidence Intervals

a confidence interval gives a range of values within which it is likely that the true population value lies.

It is defined as follows: 95% confidence interval (sample mean –

1.96 SE) to (sample mean + 1.96 SE). a 99% confidence interval (calculated as

mean ± 2.56 SE)

T-distribution

The t-distribution is similar in shape to the Normal distribution, being symmetrical and unimodal, but is generally more spread out with longer tails.

The exact shape depends on a quantity known as the ‘degrees of freedom’, which in this context is equal to the sample size minus 1.

T-distribution

One-sample t-test

Null hypothesis: Sample mean does not differ from hypothesized mean, e.g., 0 (Ho: =0)

A t-statistics (t) is calculated. t is the number of SEs that separate the sample mean from

the hypothesized value. The associated P value is obtained by comparison with the t

distribution. Larger the t-statistics, lower the probability of obtaining such a

large value, thus p is smaller and more significant.

Paired t-test

Used with paired data. Paired data arise in a number of

different situations, a matched case–control study in which

individual cases and controls are matched to each other, or

A repeat measures study in which some measurement is made on the same set of individuals on more than one occasion

Paired t-test

Two-sample t-test

Comparison of two groups with unpaired data. E.g., comparison of individuals of

treatment and those of control for a particular variable.

Now there are two independent populations thus two STDs

Calculation of pooled STD

The pooled SD for the difference in means is calculated as follows:

Calculation of pooled SE

the combined SE gives more weight to the larger sample size (if sample sizes are unequal) because this is likely to be more reliable. The pooled SD for the difference in means is calculated as follows:

Two sample T-test

Comparison of means of two groups based on a t-statistics and its student’s t-distribution. dividing the difference between the

sample means by the standard error of the difference.

T-statistic

A P value may be obtained by comparison with the t distribution on n1 + n2 – 2 degrees of freedom.

Again, the larger the t statistic, the smaller the P value will be.

Example

X-gene exprs. Tumor Control

# of samples 119 117

Mean 81 95

Std 18 19

Calculation of SD

Calculation of SE

T-statistic

t = (95-81)/2.41 = 14/2.41 = 5.81, with a corresponding P value less than 0.0001.

Reject null hypothesis that states that sample means do not differ.

Analysis of Variance

ANOVA A technique for analyzing the way in

which the mean of a variable is affected by different types and combinations of factors. E.g., the effect of three different diets on total

serum cholesterol

Sample Experiment

Variance:

Sum of squares calculations

totalwithinbetween

Degrees of freedom

Sources of variation

P value of 0.0039 means that at least two of the treatment groups are different.

Multiple Tests

Post hoc comparisons between pairs of treatments.

Overall type I error rate increases by increasing number of pairwise comparisons.

One has to maintain the 0.05 type I error rate after all of the comparisons.

Bonferroni Adjustment

0.05/#of tests Too conservative

NonParametric methods

Many statistical methods require assumptions. T-test requires samples are normally

distributed. They require transformations

Nonparametric methods require very little or no assumptions.

Wilcoxon signed rank test for paired data

Wilcoxon signed rank test

Central venous oxygen saturation on admission and after 6 h into ICU.

Take the difference between the paired data points. Patients have

SvO2 values on admission and after 6 hours.

Central venous oxygen saturation on admission and after 6 h into ICU.

Rank differences regardless of their sign.

Give a sign to the ranked differences

Calculate

Sum of positive ranks (R+)

Sum of negative ranks (R-)

Sum of positive and negative ranks

Critical values for WSR test when n = 10

5

Wilcoxon sum or Mann-Whitney test

Wilcoxon signed rank is good for paired data.

For unpaired data, wilcoxon sum test is used.

Steps of Wilcoxon rank-sum test

Total drug doses in patients with a 3 to 5 day stay in intensive care unit.

Rank all observations in the increasing order regardless of groupings

Use average rank if the values tie

Add up the ranks Select the smaller

value, calculate a p-value for it.

Critical values

Correlation and Regression

Correlation quantifies the strength of the relationship between two paired samples.

Regression expresses the relationship in the form of an equation.

Example: whether two genes, X and Y are coregulated, or the expression level of gene X can be predicted based on the expression level of gene Y.

Product moment correlation

r lies between -1 and +1

Age and urea for 20 patients in emergency unit

Scattergram

r = 0.62

Confidence intervals around r

Confidence of r

Misuse of correlation

There may be a third variable both of the variables are related to

It does not imply causation. A nonlinear relationship may exist.

Regression

Method of least squares

The regression line is obtained using the method of least squares. Any line y = a + bx that we draw through the points gives a predicted or fitted value of y for each value of x in the dataset.

For a particular value of x the vertical difference between the observed and the fitted value of y is known as the deviation or residual.

The method least squares finds the values a and b that minimizes the sum of squares of all deviations.

Age and urea level

Residuals

Method of least squares