Statistical Analysis of Dihybrid Crosses (HL) (HL IB Biology)

Use of a chi-squared test on data from dihybrid crosses

• The difference between expected and observed results in experiments can be statistically significant or insignificant (happened by chance)
• If the difference between results is statistically significant it can suggest that something else is happening in the experiment that isn’t being accounted for
  • For example, linkage between genes
• A statistical test called the chi-squared test determines whether there is a significant difference between the observed and expected results in an experiment
• The chi-squared test is completed when the data is categorical (data that can be grouped)

Calculating chi-squared values

1. Obtain the expected and observed results for the experiment
2. Calculate the difference between each set of results
3. Square each difference (as it is irrelevant whether the difference is positive or negative)
4. Divide each squared difference by the expected value and get a sum of these answers to obtain the chi-squared value

Analysing chi-squared values

• To work out what the chi-squared value means, a table that relates chi-squared values to probabilities is used
• If the chi-squared value represents a larger probability than the critical probability then it can be stated that the differences between the expected and observed results are due to chance
• If it represents a smaller probability than the critical probability then the differences in results are significant and something else may be causing the differences
• To determine the critical probability biologists generally use a probability of 0.05 (p = 0.05)
  • They allow that chance will cause five out of every 100 experiments to be different
• The number of comparisons made must also be taken into account when determining the critical probability. This is known as the degrees of freedom

• Every hypothesis test must begin with a clear null hypothesis (what we believe to already be true) and alternative hypothesis (how we believe the data pattern or probability distribution might have changed)
• The null hypothesis is denoted H₀ and sets out the assumed population parameter given that no change has happened
  • The standard null hypothesis is that there is no significant difference between the expected and observed frequencies, and any difference that does occur is due to chance
• The alternative hypothesis is denoted H₁  and sets out how we think the population parameter could have changed
  • The standard alternative hypothesis is that there is a significant difference between the expected and observed frequencies
• When a hypothesis test is carried out, the null hypothesis is assumed to be true and this assumption will either be accepted or rejected

• The expected number of each phenotype is the fraction of the total number of rabbits governed by the 9:3:3:1 ratio
• These are ⁹/₁₆, ³/₁₆, ³/₁₆ and ¹/₁₆ of 160, respectively
• In order to understand what this chi-squared value of 2.93 says about the data, a table relating chi-squared values to probability is needed

Chi-squared critical values table

• The chi-squared table displays the probabilities that the differences between expected and observed are due to chance
• The degrees of freedom can be worked out from the results. It is calculated by subtracting one from the number of classes
• In this example, there are four phenotypes which means four classes, 4 – 1 = 3
• This means that the values in the third row are important for comparison
• For this experiment, there is a critical probability of 0.05
• This means that 7.82 is the value used for comparison
• The chi-squared value from the results (2.93) is much smaller than 7.82
• 2.93 would be located somewhere to the left-hand side of the table, representing a probability much greater than 0.1
• This means that there is no significant difference between the expected and observed results, any differences that do occur are due to chance
  • We accept the null hypothesis

• The expected number of each phenotype is the fraction of the total number of flies governed by the 9:3:3:1 ratio
• These are ⁹/₁₆, ³/₁₆, ³/₁₆ and ¹/₁₆ of 597, respectively
• In order to understand what this chi-squared value of 235 says about the data, a table relating chi-squared values to probability is needed

Chi-squared critical values table

• The chi-squared table displays the probabilities that the differences between expected and observed are due to chance
• The degrees of freedom can be worked out from the results. It is calculated by subtracting one from the number of classes
• In this example, there are also four phenotypes which means four classes, 4 – 1 = 3
• This means that the values in the third row are important for comparison
• For this experiment, there is a critical probability of 0.05
• This means that 7.82 is the value used for comparison
• The chi-squared value from the results (235) is much greater than 7.82
• 235 would be located somewhere off the far right of the table, representing a very small probability, much less than 0.001
• This means that there is a significant difference between the expected and observed results, any differences that do occur are due to a factor other than chance
  • We reject the null hypothesis
• In this case, that factor is (autosomal) gene linkage

Conclusion

• The alleles for black/brown body colour and straight/curved wings are linked, ie. carried on the same chromosome (autosome)

NOS: Students should recognise that statistical testing often involves using a sample to represent a population

• When handling data involving large populations, it is often impossible to collect data on every single member of that population
• In these scenarios the best option is to take a large sample of the data from a percentage of the population
  • The larger the sample, the higher the chance that the data from the sample is representative of the whole population
• This is common in ecological sampling, such as when collecting data using a quadrat, or in the context of studying inheritance of within populations
• In many experiments the sample is the replicated or repeated measurements