Introduction to Statistics: Probability and Types of Analysis Probability and parametric / non-parametric analyses

Glossary – a reminder….. Population – all members of a group Sample – the subset that we study or measure Random sample – data is sampled randomly Variables – Factors that could influence the results Treatments – What you are testing Replication – repetition of the treatments / sampling Levels – the levels at which the treatment is applied Control – zero treatment; allows for comparisons Statistic – value calculated for a specific statistical test

Probability - Rolling a Dice The chance of something happening Six possible outcomes, so the probability of rolling a 3? And the probability of rolling an even number? All of these probabilities can be written as a decimal (0 to 1) Which leads us to p-values

The p-value The probability that the observed result is consistent with the null hypothesis (i.e. no difference, association etc). If the probability is below a predetermined level (usually 0.05) then reject the null hypothesis and accept the alternative hypothesis. p-value of 0.05 Only 5% probability (p=0.05) that our results are due to chance. Therefore a 95% probability that the difference we see is due to our experiment and not chance.

Why the magical p=0.05? It is a rather arbitrary value. A standard value. Any result with a p-value < 0.05 is said to be significant. Are these therefore significant? P=0.02 P=0.055 P=0.50

Critical Significance Values P-value Interpretation Symbol P>0.05 No evidence against null hypothesis (No significant difference) N.S. P<0.05 Moderate evidence against null hypothesis (significant) * P<0.01 Strong evidence against null hypothesis (very significant) ** P<0.001 Very strong evidence against null hypothesis (highly significant) ***

The Normal Distribution The spread of the data is symmetrical. Data is more concentrated in the middle than in the tails on either side. Typical bell shaped curve.

Confidence Limits How well do values computed from the sample reflect the trends in the entire population? Use Confidence Limits – they are applied to frequency distributions either side (+/-) of the mean, and are based around the standard deviation estimations: ONE standard deviation measure = 68 %; TWO standard deviation measures = 95 %; THREE standard deviation measures = 99 %.

1 standard deviation from the mean A normal distribution curve showing the proportion of values (%) which fall within a given distance from the mean Mean 2 standard deviations from the mean 3 standard deviations from the mean Freq. 34 % 34 % 13.5 % 13.5 % 2.25 % 2.25 % Values

An example: The Galapagos Giant Tortoise! Example: The mean life span for a Galapagos giant land tortoise is 100 years, and the standard deviation is 30. There is a 34 % probability that other tortoises of the same species will be 70 to 100 years and 34 % probability they will be 100 to 130 years (68%, 70-130). There is a 95 % probability that other tortoises of the same species will be more than 40 years or less than 160 years; There is a 0.5 % probability that other tortoises of the same species will be less than 10 years or more than 190 years.

Standard Error of Differences Between Means – the yardstick for deciding whether things are statistically ‘significant’. Differences of up to 1 s.e.d.m between 2 samples would arise quite frequently by chance (up to 68%). But differences exceeding 2 s.e.d.m would occur only 5% by chance – therefore it is unlikely (this is where the critical probability of 0.05 comes from). In other words, we could conclude that the populations on test have different means. At this stage, differences can be accepted as statistically different.

Statistical Testing Various statistical tests are available Check the data meets the assumptions required by the test Calculate the test statistic Determine the probability or p-value Reject or accept your null hypothesis

P=0.05 P=0.01

One or two tailed??? One-tailed test Two-tailed test If we can predict an If we just predict some increase or a decrease kind of change If test statistic lies in the ‘red zone’ we reject the null hypothesis

Types of Statistical Analysis Descriptive statistics Non-parametric statistics Parametric statistics

Parametric Statistics Each test has certain parameters (assumptions) that the data must meet For example Must follow a known distribution (usually normal) Each dataset must have an equal variance If the data does not meet these parameters our results may result in errors More powerful that non-parametric methods

Some other distributions Skewed to the left Bimodal

Non-Parametric Statistics No particular distribution of the data needed ‘Distribution free’ Less powerful than parametric tests ‘Safer’ Cannot answer complicated questions e.g.: complex interactions between variable, modelling associations etc.

When to Use What? Computer software such as Minitab, Genstat and to some extent MS Excel mean that we don’t have to learn the complex maths! But it is important that we use the computer to do the right test

Question Test Parametric Non-parametric Is there a relationship between variables? Correlation test Pearson Spearman Is there a difference between independent groups? Independent measures, 2 groups Independent-measures t-test Mann-Whitney test Independent measures, > 2 groups One-way, independent-measures ANOVA Kruskal-Wallis test Is there a difference between dependent groups? Repeated measures, 2 conditions Matched-pair t-test Wilcoxon test Repeated measures, >2 conditions One-way, repeated measures ANOVA Friedman's test

References Dytham C. 2003. Choosing and Using Statistics: A Biologist’s Guide. Oxford: Blackwell publishing. The best for statistical computing. Van Emden, HF. 2008. Statistics for Terrified Biologists. Malden, MA: Blackwell Publishing. The best for understanding how it all works. Rowntree, D. (1981) Statistics without Tears. Penguin Excellent little book and only a pound from ebay! Morris, T.R. (1999) Experimental Design and Analysis in Animal Sciences. CAB International. Townsend, J. (2002) Practical Statistics for Environmental and Biological Scientists. Wiley. Zar, J.H. (1984) Biostatistical Analysis. 2nd Edition. Prentice Hall. Statistics books in the LRC at 519.5 Minitab guidebooks at 005.379

Some useful websites http://www.mathsrevision.net/alevel/statistics/index.php http://www.bbc.co.uk/skillswise/numbers/handlingdata/ http://www.statsoft.com/textbook/stathome.html http://changingminds.org/explanations/research/analysis/parametric_non-parametric.htm

Are you able to … Understand what a p-value is and describe what probability is? Explore parametric and non-parametric analyses? Understand why parametric and non-parametric analyses are required? Review examples of parametric analyses and their non-parametric equivalent?

Testing for Normality on Minitab Stat – Basic Statistics – Normality test If p < 0.05 then the data is not normally distributed. This one is!