Statistics for A2 Biology

A2 statistics page 1

Statistics for A2 Biology In A2 biology we need to learn how to use statistics to analyse experimental results. Sometimes Sometimes experimental results are not very clear and so it is difficult to make a firm conclusion. In these cases an appropriate statistical test can help to clarify the results so that a valid conclusion can be made. Statistics can therefore be used to extract the maximum amount of information from experimental data, and are an essential tool of experimental biology. There are three stages in using a statistical test: choosing a test; carrying it out; and making a conclusion. Before we choose a test though, we need to learn a little about different kinds of test and different kinds of data.

What kinds of statistics tests are there? Statistics D e s c r ip tiv e S t a tis tic s R e t u r n a v a l u e t h a t s u m m a r i z es es d a t a e . g . m e a n , m e d i a n , s t a n d a r d d e v i a t io io n population index, etc.

I n f e re n tia l S t a tis tic s R e t u r n a p r ob ob a b i lili ty ty ( P - v a lu lu e ) t h at at a null hypothesis is correct.

C o m pa p a r a t iv e S t a t i s t i c s

A s s o c ia t io n S t a tis tic s

Tests for for a difference between tw o or mo re sets of data 2 e.g. t-test, χ -test, etc.

Tests for as association between two sets of data e.g. correlation, regression.

Descriptive statistics are used to summarise data so you can simplify them and plot a graph. This is what you did in AS, and includes the mean and 95% confidence interval. For non-normal data you may instead have to calculate the median and interquartile range. Inferential Statistics test a statement called the null hypothesis, and return a probability (called a P-value) that the null hypothesis is true. The null hypothesis is a mathematical statement statement and is fixed for a given test (the (the exact null hypothesis is given for each test on the next few pages). It has nothing to do with (and can be quite different from) any scientific hypothesis you may be making making about the result of the experiment. The lower the probability, the less likely it is that the null hypothesis is true, and in biology we usually take 0.05 (or 5%) as the cut-off. This may seem very low, but it reflects the facts that biology experiments are expected to produce quite varied results. • If P < 5% (0.05) then we reject the null hypothesis, and conclude that there is a significant difference or association. • If P ≥ 5% (0.05) (0.05 ) then we accept the null hypothesis, and conclude that there is no significant difference or association.

There are two kinds of inferential statistics: Comparative statistics are used to compare different sets of data to see if they are different (e.g. is this group bigger than that group?). The null hypothesis states that there is no difference between the sets of data. You must also choose between matched-sample experiments and independent sample experiments. In a matched-sample experiment each measurement from the first group matches up with corresponding measurement in the other groups, perhaps because they were all made on the same subject (e.g. a " before and after" experiment). Otherwise Otherwise you have an independent sample experiment. If it isn't obvious which value matches with which, then it's probably not matched. Association statistics are used to look for an association (or correlation) between two sets of data (e.g. if this goes up does that go up?). The null hypothesis states that there is no association between the sets. A scatter graph (or mosaic chart for categoric data) of one factor against the other (without a line of best fit) indicates the association. If both factors increase together then there is a positive correlation; if one factor decreases when the other increases then there is a negative correlation; and if the scatter graph has apparently random points then there is no corr elation:

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P o sitiv e C o rre la tio n

N e g a tiv e C o rre latio n

N o C o rre la tio n

2

2

2

e l b a i r a v

e l b a i r a v

e l b a i r a v

variable 1

variable 1

variable 1

The association is described by two statistics: • First there is a P-value, which tests the null hypothesis that there is no correlation. If P<5% then the null hypothesis is rejected and there is a significant correlation (with the strength indicated by the correlation coefficient); but if P>5% then we accept the null hypothesis that there is no correlation (and the value of the correlation coefficient is meaningless). • Second there is a correlation coefficient, which gives the strength of the association. It varies from 0 (no correlation) to 1 (perfect correlation). Positive values indicate a positive correlation while negative values indicate a negative correlation. The larger the absolute value (positive or negative), the stronger the correlation. A correlation does not necessarily mean that there is a causal relationship between the factors (i.e. changes in one factor cause the changes in the other). The changes may both be caused by a third factor, or it could be just coincidence. Further controlled studies would be needed to find out.

What kinds of data are there? There are different statistical tests designed for different kinds of data. Data can be classified as quantitative (numbers) or qualitative (words).

Data Q uantitative Data (numbers) normally distributed

Q ualitative Data (words)

n o t n o r m a l ly distributed or ordinal

can be ranked

N o rm a lly D is tr ib u ted D a ta

O rd in a l D a ta

u se p ara m e tric te sts

u se n o n p ara m e tric te sts

cannot be ranked

N o m in a l D a ta u se freq u en c y tests

• Most quantitative measurements (e.g. length, mass, temperature, rates, counts) are normally-distributed, especially if you

have a large number of repeats. For normally-distributed data you can use the most powerful Parametric Tests. • Other quantitative measurements are not normally-distributed. These include arbitrary scales like "1-5", calculated data

and data sets with extreme outliers. In this case the parametric tests are invalid, so choose Non-Parametric Tests. • Some qualitative data can be ranked, so can be replaced by numerical ranks (e.g. big, medium, small become 1, 2, 3).

These can then be analysed using Non-Parametric Tests. • Finally, the data can simply be categories that cannot be ranked ( e.g. colours, shapes, species). We can't do maths on

categoric data, but we can count the numbers in each category to give frequencies, and then compare these observed frequencies with some expected frequencies using Frequency Tests.

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All the different statistical tests you will come across are summarised in this table. Don't panic, you don't need to learn this!

Descriptive Statistics

Summarise Data Chart

Comparative Statistics

Independent Samples

2 groups ≥ 2 groups

2 groups

Matched Samples

≥ 2 groups

Chart Association Statistics

Test for an Association Details of Linear Relationship

Parametric Tests (for normal data)

Non-Parametric Tests (for ordinal data)

Frequency Tests (for nominal data)

Mean, Standard Deviation, 95% CI

Median, Quartiles

-

Bar Chart

Box Plot

Pie or Bar Chart

Unpaired t-Test

Mann-Whitney U-Test

Anova

Kruskal-Wallis Test

Paired t-Test

Wilcoxon Test

Matched Anova

Friedman Test

Scatter Graph

Scatter Graph

Mosaic Chart

Pearson Correlation

Spearman Correlation

Chi-Squared Test of Association

Linear Regression

-

-

Chi-Squared Test or G-Test

The Six Steps in a Statistics-Based Investigation 1. Choose a Test

While planning your investigation, choose a suitable statistical test using Merlin's "Choose a Test" or the table above.

2. State the Null Hypothesis

Look up the test in this document. State the null hypothesis as precisely as you can for your experiment; e.g. "There is no difference between the means of the plant heights in the two areas". This statement is what the statistical test will actually test.

3. Obtain Results

Carry out the investigation and obtain results. For hypothesis-testing we need as many replicate measurements as possible, and as a guide, aim for at least 10 replicates and preferably 20 in each set.

4. Present the Data

Present your raw data in a neat results table. Use Merlin to calculate descriptive statistics, like mean and 95% CI, and plot an appropriate graph.

5. Carry out the Type the Merlin formula for your chosen test into an empty cell e.g. =TT ESTP(B3:B12,C3:C12). Statistical Test This will return the P-value for that test. It's a good idea to format this cell as a percentage (Format menu > Cells > Number tab > Percentage), so for example a P-value of 0.02 appears as 2%.

6. Make a Conclusion

State whether you accept or reject the null hypothesis, and write a sentence explaining exactly what that means in this case. For example if P was < 5% then you would reject the null hypothesis and say that the plants in group A are significantly taller than the plants in group B. Or if P was > 5% then you would accept the null hypothesis and say that as far as you can tell from the data, there is no significant difference between the heights of the two groups of plants. The wording of the conclusion is important, so use these examples as a guide and think carefully about what you are saying. Note that if P > 5% we haven't proved the null hypothesis, but since our data are consistent with it, we accept it.

There are lots of examples of conclusions on the next five pages. This is a statistics reference guide, which describes each of the tests in detail.

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Unpaired t-Test (t-test) This test is used to compare two sets of data, and it tests the null hypothesis that the two sets have the same mean. The data must be normally-distributed and there must at least 10 replicates (and preferably much more). If P<5% then the null hypothesis is rejected and there is a significant difference between the two means. The Merlin function is =TTESTU(range1, range2). In this example the effect of two fertilisers on yield of potatoes is compared. The formula =TTESTU(B3:B12,C3:C12) is typed into cell B15 and formatted as a percentage. The P value is >5% so we accept the null hypothesis and conclude that there is no significant difference between the two fertilisers.

Paired t-Test This test is used to compare two sets of paired (matched) data, and it tests the null hypothesis that the mean difference between the pairs is zero. The data must be in pairs, they must be normally-distributed and there must at least 10 replicates (and preferably much more). If P<5% then the null hypothesis is rejected and there is a significant difference between the two sets. The Merlin function is =TTESTP(range1, range2). This example compares pulse rate before and after eating a large meal. Because each individual had their pulse measured before and after the meal then the data are paired. The formula =TTESTP(B3:B12,C3:C12) is typed into cell B15 and formatted as a percentage. The P value is <5% so we reject the null hypothesis and conclude that pulse rate is significantly higher after a meal.

ANOVA (analysis of variance) This test is used to compare two or more sets of data, and it tests the null hypothesis that the sets have the same mean. The data must be normally-distributed and there must at least 10 replicates (and preferably much more). If P<5% then the null hypothesis is rejected, and at least one of the sets is significantly different. The Merlin function is =ANOVA(range), where each column in the range is a different set. This example compares 3 different colours of light on the rate of photosynthesis in Elodea measured by length of oxygen bubble produced in a given time. The formula =ANOVA(B3:D12) is typed into cell B15 and formatted as a percentage. The P value is <5% so we reject the null hypothesis and conclude that at least one of the colours is significantly different. From the means, green must be significantly lower than the others.

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Matched Samples ANOVA This test is used to compare two or more sets of matched data, and it tests the null hypothesis that the mean difference between the sets is zero. The data must be matched, they must be normally-distributed and there must at least 10 replicates (and preferably much more). If P<5% then the null hypothesis is rejected and at least one of the sets is significantly different. The Merlin function is =ANOVAM(range), where each column in the range is a different set. This example compares the mass of food eaten by 8 deer at four different times of year. The formula =ANOVAM(B3:E10) is typed into cell B13 and formatted as a percentage. The P value is <5% so we reject the null hypothesis and conclude that at least one of the months is significantly different. From the means, significantly less food is eaten in May and August.

Mann-Whitney U-Test This test is used to compare two sets of data, and it tests the null hypothesis that the two sets have the same median. The data can be any form so long as they can be ranked and there must at least 10 replicates (and preferably much more). If P<5% then the null hypothesis is rejected and there is a significant difference between the two medians. The Merlin function is =UTEST (range1, range2). This example compares the abundance of blown algae (measured on a 1-5 score) on two different shores. T he formula =UTEST(B3:B12,C3:C12) is typed into cell B13 and formatted as a percentage. The P value is <5% (just!) so we reject the null hypothesis and conclude that there is significantly more algae on the sheltered shore.

Wilcoxon Matched-Pairs Test This test is used to compare two sets of paired (matched) data, and tests the null hypothesis that the median difference between the pairs is zero. The data must be in pairs; can be any form that can be ranked and there must at least 10 replicates (and preferably much more). If P<5% then the null hypothesis is rejected and there is a significant difference between the two sets. The Merlin function is =WILCOXON(range1, range2). This example compares memory scores (number of words recalled from a text) before and after drinking alcohol. The formula =WILCOXON(B3:B12,C3:C12) is typed into cell C13 and formatted as a percentage. The P value is <5% so we reject the null hypothesis and conclude that memory is significantly worse after drinking alcohol.

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Kruskall-Wallace Test This test is used to compare two or more sets of data, and it tests the null hypothesis that the sets have the same median. The data can be any form that can be ranked and there must at least 10 replicates (and preferably much more). If P<5% then the null hypothesis is rejected and there is a significant difference between at least one of the sets. The Merlin function is =KWTEST(range), where each column in the range is a different set. This example compares decay rates of three species of leaf, measured by % of leaf area remaining after 8 weeks burial. The formula =KWTEST(B3:D11) is typed into cell B12 and formatted as a percentage. The P value is <5% so we reject the null hypothesis and conclude that there is a significant difference between at least two of the leaves. From the box plot, the beech leaves must decay significantly more slowly than the other two species.

Friedman Test This test is used to compare two or more sets of matched data, and it tests the null hypothesis that the median difference between the sets is zero. The data can be any form that can be ranked but must be matched, and there must at least 10 replicates (and preferably much more). If P<5% then the null hypothesis is rejected and at least one of the sets is significantly different. The Merlin function is =FRIEDMAN(range), where each column in the range is a different set. This example compares the symptoms of patients (on a score system) before and after treatment with a drug. The formula =FRIEDMAN(B3:D12) is typed into cell B13 and formatted as a percentage. The P value is <5% so we reject the null hypothesis and conclude that at least one of days is significantly different. From the medians, there is a significant drop in symptoms after treatment, so the drug works.

Chisquared (χ2) Test This test is used for frequencies of categoric data, and it tests the null hypothesis that there is no difference between the observed and expected frequencies. If P<5% then the null hypothesis is rejected and there is a significant difference between the frequencies. The Excel function is =CHITEST(obsrange, exprange). There are different ways of calculating the expected frequencies: Sometimes the expected frequencies can be calculated from a quantitative theory such as Mendel's laws of genetics. In this example the frequencies of flower colours from a genetic cross are compared to an expected 3:1 ratio. The expected frequencies can be calculated from the total number of observations (929) using simple Excel formulae. The formula =CHITEST(B2:B3,C2:C3) is typed into cell C5 and formatted as a percentage. The P value is >5% so we accept the null hypothesis and conclude that the observed data are consistent with Mendel's law.

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Other times the expected frequencies can be calculated by assuming that the frequencies in all the categories should be the same. In this example the frequencies of boys and girls born in a hospital over a period of time are compared to an expected 1:1 ratio. The expected frequencies can be calculated from the total number of observations (445) using simple Excel formulae. The formula =CHITEST(B2:B3,C2:C3) is typed into cell C5 and formatted as a percentage. The P value is >5% so we accept the null hypothesis and conclude that there is no difference between male and female births.

G-Test This test is also used for frequencies of categoric data, and it can be used whenever the chisquared test can be used. Many statisticians prefer the G-test to the chisquared test. It has the same null hypothesis as the chisquared test. The Merlin function is =GTEST(obsrange, exprange). For the genetic cross example above GTEST gives the P-value 53.04%. For the sex of baby example above GTEST gives the P-value 6.45%. So in both cases the conclusion is the same.

Pearson Linear Correlation This test is used to test for a correlation between two sets of normally-distributed data. The correlation coefficient is called r and is calculated using the function =PEARSON(range1, range2). The corresponding P-value, which tests the null hypothesis that r=0, is calculated using the function =PEARSONP(range1, range2). In this example the heights of 10 fathers are compared with their sons. A scatter graph is plotted and the formula =PEARSONP(B2:B12,C2:C12) is typed into cell C13 and formatted as a percentage and the formula =PEARSON(B2:B12,C2:C12) is typed into cell C14. The P value is <5% so the null hypothesis is rejected and we conclude that there is a significant positive correlation with a strength of 0.72, so tall fathers do have tall sons.

Spearman Rank Correlation This test is used to test for a correlation between two sets of non-normal data. The correlation coefficient is called rs and is calculated using the function =SPEARMAN(range1, range2). The corresponding Pvalue, which tests the null hypothesis that r s=0, is calculated using the function =SPEARMANP(range1, range2). In this example the social status of hens (measured by their pecking order) is compared with their mass. A scatter graph is plotted and the formula =SPEARMANP(B2:B11,C2:C11) is typed into cell C12 and formatted as a percentage and the formula =SPEARMAN(B2:B11,C2:C11) is typed into cell C13. The P value is <5% so the null hypothesis is rejected and we conclude that there is a significant negative correlation with a strength of -0.77, so big hens are higher up the pecking order.

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Chisquared Test of Association This test is used to test for an association between two factors that are measured by categoric data. The two sets of categories are the rows and columns of a contingency table, which contains the frequencies. The association coefficient is called Cramer's V and is calculated using the Merlin function =CRAMER(range). The corresponding P-value is 2 called the χ test of association and it tests the null hypothesis that there is no association (i.e. Cramer's V=0). It is calculated using the Merlin function =CHIASSOC(range). This example investigates whether there are more nests in birch or sycamore trees, in other words whether there is an association between tree species and birds' nests. The formula =CHASSOC(B2:C3) is typed into cell B4 and formatted as a percentage and the formula =CRAMER(B2:C3) is typed into cell B5. The P value is <5% so the null hypothesis is rejected and we conclude that there is a significant association, though it is only weak, with a strength of 0.22.

Linear Regression This is used to describe a linear relationship between two sets of data. It is used when you already know that one variable causes the changes in the other variable (i.e. there is a causal relationship). Regression fits a straight line to the data, and gives the values of the intercept and slope (or gradient) of that line ( a and b in the equation y = a + bx). The Merlin function =REGRESS(xrange, yrange, flag) returns values for the slope and intercept as well as their 95% confidence intervals. The flag value determines whether the intercept is fixed at zero. REGRESS is an array function, so a square of 4 cells is selected and the function is entered with ctrl-shiftEnter. In this example the absorption of a yeast cell suspension is plotted against its cell concentration from a cell counter. The formula =REGRESS(A2:A12,B2:B12,0) was typed into cells b15:c16 and entered as an arr ay formula. The intercept was fixed at zero because 0 cells have 0 absorbance. The straight trendline was also plotted on the scatter graph. The regression can then be used to make quantitative predictions. For example, we could predict that a sample with an absorbance of 1.37 has a 7 -3 cell concentration of 9 x 10 cells cm .

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Excel Tips •

Take time to tidy all results tables, as Excel's default formatting isn't very good e.g. line up titles with values. Format all numbers to an appropriate number of decimal places (Format menu > Cells > Number tab > Number).

•

Format P-values as percentages (Format menu > Cells > Number tab > Percentage). T his automatically multiplies the

•

•

•

P-value by 100 and adds the % sign to make small P values easier to read and understand. Use Merlin for charts, even if Excel provides the same chart (e.g. a scatter graph). T he format of Merlin charts is better for scientific data. Even so, take time to tidy up all charts, adjusting the size, shape, colour, font size, etc. to make the chart clear. Never use an Excel line chart.

Name That Test! For each of these investigations, choose the best statistics test. In some cases there may be more than one equally good answer, generally if you can't tell whether the dependent variable is normally distributed or not. 1.

The leaf area of many plants (of the same species) growing in open and shaded areas is measured to investigate the effect of light intensity on leaf area.

2.

The number of mayfly larvae found in the middle and edge of a stream was compared at 12 stations along the stream.

3.

4.

5. 6.

To test whether the γ -radiation used to sterilise barley seeds affected the seeds themselves, batches of 100 seeds were exposed to three different doses of radiation (none/low/high) and then planted to see if the seeds germinated or not. Thirteen plants were kept in large sealed glass bottles and the carbon dioxide concentration inside was measured over a period of 24h to see if the decrease in carbon dioxide per hour during the day was different from the increase in carbon dioxide per hour during the night. What is the relation between planting density of oat seedlings and yield of seeds per plant? Is there a relation between the number of seeds produced by a plant and mean mass per seed?

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Doctors were alarmed by the apparently high incidence of a childhood disease in their town Given the known national incidence of the disease, how can they tell if the frequency in their town is different?

8.

Investigating the abundance of orchids in four different fields.

9.

Investigating the effect of three different diets on Body Mass Index .

10. A student using lichens as a measure of air pollution measured lichen diversity at several locations in a town to see if there was a relation between distance from town center and diversity index. 11. In order to find the best preparation of a vaccine, three different forms of the vaccine were inject into the same 20 healthy volunteers on three separate occasions, and the concentrations of antibodies in the blood was measured. 12. To investigate the effect of handling by humans on lab rats, the activity of 10 handled and 10 unhandled rats was scored on a 1-5 scale (quiet to very active).

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Statistics for A2 Biology

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