introduction to statistics alastair kerr, phd. think about these statements (discuss at end)...
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Introduction to Statistics
Alastair Kerr, PhD
Think about these statements (discuss at end)
Paraphrased from real conversations:– “We used a t-test to compare our samples”– “These genes are the most highly expressed in my
experiment: this must be significant”– “No significant difference between these samples
therefore the samples are the same”– “Yes I have replicates, I ran the same sample 3 times”– “We ignored those points, they are obviously wrong!”– “ X and Y are related as the p-value is 1e-168!”– “I need you to show this data is significant”
Basic Probability
Which of these sequence of numbers is random? (outcomes 0 or 1, unsorted data)
1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Binomial Distribution
Thought experiment Everyone flip a coin 10 times and count
number of ‘heads’ Most frequent observation? Least? Pattern of observations between these?
How would these factors affect the graph shape? Using a dice instead of a coin and looking for the
number 6? Increasing the number of times the coin was
flipped?
Binomial Distributions
Types of Data
Discrete or Continuous Discrete: values for a finite number of samples Continuous: infinite population...
Parametric or Non-parametric Fits a known distribution Fits specific properties Specific tests are available if and only if the data
is parametric
Normal Distribution
the curve has a single peak the mean (average) lies at the centre of the
distribution distribution is symmetrical around the mean the two tails of the distribution extend
indefinitely and never touch the horizontal axis (continuous distribution)
the shape of the distribution is determined by its Mean (µ) and Standard Deviation (σ).
Variance and standard deviation
Variance is just how dispersed your data is from the mean.
Formalised: "The average of the square of the distance of
each data point from the mean" Standard deviation is the square root of the
variance aka RMS [or root mean squared] deviation Really just the distance to the mean from a
‘average’ sample
Normal distribution
95% of the data are within 2σ [standard deviation] of the mean
aka the 95% confidence interval
Understanding 'average'
When talking about average or mean, we commonly refer to the arithmetic mean. sum of samples / number of samples
Other Pythagorean means: geometric and harmonic
geometric mean – average of factors harmonic mean – average of rates
Other ways to describe Mode - most common value Median – central value in an ordered list
of numbers
When geometric mean is useful
nth root of the product of n numbers Or mean of the log values of a dataset,
converted back to base10 Factors such as ratio microarray data
e.g. for 'fold change' or other non-linear proportions less sensitive to extremely large values, it can be applied to data with relatively large fluctuations.
When harmonic mean is useful
Mean of the reciprocal of values, then take the reciprocal again to convert back.
Looking at ‘rates of change’ I’ve used it for the rate of change of nucleotide
substitutions Gives the lowest values of all the means Good way for limiting the effect of outliers (if outliers
are all large values…)
Why use median?
Remember median is the central value of a ranked list
What is the median of <pick 5 numbers> Great to use for skewed distributions Similar to the mean in a normal distribution
Why? Cannot really use SD or variance – instead
quartiles and interquartile range [IQR]
Quartiles and Quantiles
Quantiles are points taken at regular intervals on a ranked list of data
The 100-quantiles are called percentiles. The 10-quantiles are called deciles. The 5-quantiles are called quintiles. The 4-quantiles are called quartiles.
Quartiles 'middle 50', or inter quartile range [IQR] = 1st to 3rd quartile first quartile (lower quartile)
cuts off lowest 25% of data = 25th percentile second quartile (median)
cuts data set in half = 50th percentile third quartile (upper quartile)
cuts off highest 25% of data, or lowest 75% = 75th percentile
Visualisation: boxplot
aka candlestick box = 50% of data whisker =lines dots = outliers
Easy way to visualise the properties of multiple distributions beside each other
Visualisation: Cumulative Distribution Function
How does this CDF differ?
Hypothesis testing
Define your question Bad: “Is this significant?”
You need to compare to a model, usually that model is random chance
Good: “Does this data differ significantly from random chance compared to this other set?”
Hypothesis testing
Test a hypothesis NOT a result Bad: Gene XYZ is the most expressed in our
data set, is it significant? Ok to get hypothesis to test from eye-balling data, but
define on a biological concept, not a cherry-picked data point
OK to use to build a hypothesis: cold shock protein cspC is the most expressed gene, does this experiment enrich for cold shock proteins?
OK if enough REPLICATES
Hypothesis testing
'Bayesian' analysis– model testing against is not random – Instead 'Priors” exist, knowledge of the
system– Examples
• The 3 envelope puzzle• Odds at racing
Hypothesis testing
Test if parametric by using a non-parametric test against the normal distribution – e.g. Shapiro-Wilk or Anderson-Darling test
Question: are samples A and B different? Null hypothesis What is the likelihood that
differences between A and B are from random chance
You are testing ONE hypothesis. If it does not pass, the inverse question is not necessarily true
Testing 2 groups
If Normal Distribution Analysis of variance
[ANOVA] e.g. t-test
Most powerful tests to use but data MUST resemble parametric
If Non-Parametric KS [Kolmogorov-Smirnov]
test (Q-Q testing) Mann-Whitney (rank sum) Chi-squared
Fishers exact test if small numbers
Test if parametric by using a non-parametric test Test if parametric by using a non-parametric test against the normal distribution – e.g. Shapiro-Wilk against the normal distribution – e.g. Shapiro-Wilk or Anderson-Darling testor Anderson-Darling test
P-values: multiple testing
P-values:Correlation & Causation
Replicates
• Your statement about your data is limited by what you tested by replication.
– It may be significant but for different reasons that you think
• Replicates show the noise in the system: but what system?
– Technical, each experimental unit• Machine Variance
– Pipetting variance, Temperature Variance...• Biological: Changes in what you are
examining.– from person to person, cell to cell, grown
condition to growth condition
Define the Number of Biological Repeats
Discuss the problems with each of these
“We used a t-test to compare our samples”
“These genes are the most highly expressed in my experiment: this must be significant”
“No significant difference between these samples therefore the samples are the same”
“Yes I have replicates, I ran the same sample 3 times”
“We ignored those points, they are obviously wrong!”
“ X and Y are related as the p-value is 1e-168!”
“I need you to show this data is significant”