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Marshall University School of Medicine Department of Biochemistry and Microbiology BMS 617 Lecture 6 – Multiple comparisons, non-normality, outliers Marshall University Genomics Core Facility Analyzing data without a plan • The framework for hypothesis testing assumes all aspects of the experimental design are defined before the experiment and the analysis are performed – Not doing this can invalidate the interpretation of a p-value – Easy trap to fall into • This happens a lot! Marshall University School of Medicine Multiple comparisons • Previous lecture discussed multiple comparisons and their effect on interpretation of p-values – In that context, the multiple comparisons were part of the experimental design • In this case, can correct for them or otherwise analyze the data appropriately • Not having a complete plan can introduce multiple comparison effects in an uncontrolled manner Marshall University School of Medicine Examples of multiple comparisons • Trying multiple statistical tests for the same data set – “A t-test doesn’t give me significance… let’s separate into three groups instead and try an ANOVA” • Trying multiple algorithmic implementations of the same test • In multiple regression (we will discuss this later…), choosing to include or exclude different independent variables – For complex data sets, trying enough approaches will almost always result in a “statistically significant” result Marshall University School of Medicine Sequential Analyses • Another common approach is to try an experiment, and if the result is not statistically significant, to then repeat it with additional samples or experimental replicates – Another form of multiple comparisons – Problem with this approach is that it is biased towards a statistically significant result • Stop experimenting if a result is statistically significant • Continue experimenting otherwise • In theory you can always get a statistically significant result with this approach – Though it may take a very long time… Marshall University School of Medicine Publication Bias • Remember, the interpretation of a p-value is the probability of observing data at least as extreme as the data observed, assuming the null hypothesis is true – This is not the same as the probability the null hypothesis is true • Most p-values we see are in journal articles • There is a strong preference to publish results which are “statistically significant,” i.e. which have p < 0.05 • Some of these results are “real” and some are false positives • Because the publications are selected based on the p-value, the interpretation of a p-value in published results is skewed – If we assume the null hypothesis is true, and the result was published, the probability of a false positive can be much higher than 5% Marshall University School of Medicine Normally distributed data • Many of the statistical tests we will study rely on the assumption that the data were sampled from a normal distribution • How reasonable is this assumption? • The normal distribution is an ideal distribution that likely never exists in reality – Includes arbitrarily large values and arbitrarily small (negative) values • However, simulations show that most tests that rely on the assumption of normality are robust to deviations from the normal distribution Marshall University School of Medicine The ideal normal distribution • Image shows data sampled from a theoretical normal distribution • Uses a very large sample size • Close approximation to theoretical distribution Marshall University School of Medicine Samples from a normal distribution Marshall University School of Medicine Tests for normality • It is possible to perform tests to see if the sample data are consistent with the assumption that they were sampled from a normal distribution – Unfortunately, this is not what we really want to know… – Would really like to know if the distribution is close enough to normal for the test we use to be useful Marshall University School of Medicine Tests for normality • A test for normality is a statistical test for which the null hypothesis is The data were sampled from a normal distribution • Common normality tests include – D’Agostino-Pearson omnibus K2 normality test – Shapiro-Wilk test – Kolmogorov-Smirnov test Marshall University School of Medicine D’Agostino-Pearson omnibus K2 normality test • The D’Agostino-Pearson omnibus K2 normality test works by computing two values for the data set: – The skewness, which measures how far the data is from being symmetric – The kurtosis, which measures how sharply peaked the data is • The test then combines these to a single value that describes how far from normal the data appear to lie – Computes a p-value for this combined value Marshall University School of Medicine Problem with normality tests • If the p-value for a normality test is small, the interpretation is: – If the data were sampled from an ideal normal distribution, it is unlikely the sample would be this skewed and/or kurtotic • If the p-value for a normality test is large, then the data are not inconsistent with being sampled from a normal distribution • However… – If the sample size is large, it is possible to get a small p-value even for small deviations from the normal distribution • Data are likely sampled from a distribution that is close to, but not exactly, normal – If the sample size is small, it is possible to get a large p-value even if the underlying distribution is far from normal • Data do not provide sufficient evidence to reject the null hypothesis… – Useful to examine the values for skewness and kurtosis as well as the p-value Marshall University School of Medicine Skewness and kurtosis Marshall University School of Medicine Interpreting skewness and kurtosis • The real question we would like to answer is – How much skewness and kurtosis are acceptable? – Difficult to answer… • In general, interpret a skewness between -0.5 and 0.5 as being approximately symmetric – Between -1.0 and -0.5, or 0.5 and 1.0 is moderately skewed – Less than -1.0 or more than 1.0 is highly skewed • For kurtosis, values between -2 and 2 are generally accepted as being “within limits” – Outside this is evidence the distribution is far from normal Marshall University School of Medicine What to do if the data fail a test for normality • If the data fail a test for normality, the following options are available – Can the data be transformed to data that come from a normal distribution? • For example, if the data are negatively skewed, transforming to logs may give normally distributed data – Are there a small number of outliers that are causing the data to fail a normality test? • Next section discusses outliers – Is the departure from normality small? I.e. are the skewness and kurtosis “small”. If so, your statistical tests may still be accurate enough – Use a test that does not assume a normal distribution (a nonparametric test) Marshall University School of Medicine Non-parametric tests • The most common statistical tests assume the data are sampled from a normal distribution – T-tests, ANOVA, Pearson correlation, etc • Some other tests do not make this assumption – Mann-Whitney test, Kruskal-Wallis test, Spearman correlation, etc • However, these tests have (much) lower statistical power than their parametric equivalents when the data are normally distributed Marshall University School of Medicine Choosing non-parametric tests • When running a series of similar experiments, all data should be analyzed the same way – Use normality tests to choose the statistical test for all experiments together – Following “common practice” is acceptable… – Ideally, run one experiment just to determine whether the data look like they come from a normal distribution • For small data sets – A test for normality does not tell you much • Not likely to get a small p-value anyway – Violations of the normality assumption are more egregious – Non-parametric tests have very low statistical power Marshall University School of Medicine Outliers • Outliers are values in the data that are “far” from the other values • Occur for several reasons: – Invalid data entry – Experimental mistakes – Random chance • In any distribution, some values are far from the others – In a normal distribution, these values are rarer, but still exist – Biological diversity • If your samples are from patient or animal samples, the outlier may be “correct” and due to biological diversity – May be an interesting finding! – Wrong assumptions • For example, in a lognormal distribution, some values are far from the others Marshall University School of Medicine Why test for outliers • Presence of erroneous outliers, or assuming the wrong distribution, can introduce spurious results or mask real results • Trying to detect outliers without a test can be problematic – We tend to want to observe patterns in data – Anything that appears to be counter to these patterns seems to be an outlier – We tend to see too many outliers Marshall University School of Medicine Before testing for outliers • Before testing for outliers: – Check the data entry • Errors here can often be fixed – Were there problems with the experiment? • If errors were observed during the experiment, remove data associated with those errors • Many experimental protocols have quality control measures – Is it possible your data is not normally distributed • Most outlier tests assume the (non-outlier) data is normally distributed – Was there anything different about any of the samples • Was one of the mice phenotypically different, etc? Marshall University School of Medicine Outlier tests • After addressing the concerns on the previous slide, if you still suspect an outlier you can run an outlier test • Outlier tests answer the following question: If the data were sampled from a normal distribution, what is the chance of observing one value as far from the others as is in the observed data? Marshall University School of Medicine Results of an outlier test • If an outlier test results in a small p-value, then the conclusion is that the outlying value is (probably) not from the same distribution as the other values – Justifies excluding it from the analysis • If the outlier test results in a high p-value, there is no evidence the value came from a different distribution – Doesn’t prove it did come from the same distribution, just that there is no strong evidence to the contrary Marshall University School of Medicine Guidelines on removing outliers • If you address all the previous concerns, and an outlier test gives strong evidence of an outlier, then it is legitimate to remove it from the analysis – The rules for eliminating outliers should be established before you generate the data – You should report the number of outliers removed and the rationale for doing so in any publication using the data Marshall University School of Medicine How outlier tests work • Outlier tests work by computing the difference between the extreme value and some measure of central tendency • That value is typically divided by a measure of the variability • Resulting ratio is compared with a table or expected distribution of those values Marshall University School of Medicine Grubb’s outlier test • Grubb’s outlier test calculates the difference between the extreme value and the mean of all values (including the extreme value), and divides by the standard deviation • Resulting value is then compared to a table of critical values – Critical value depends on the sample size – If the value is larger than the critical value, then the extreme value can be considered an outlier Marshall University School of Medicine