Data Analysis

1. Data Analysis E3: Lecture 8

2. Data Analysis • Lecture Outline • Processing and Visualizing Data • Why do we do this? • Processing the Luria-Delbruck Data • Processing the Public Goods Data • Analyzing Data (using Excel) • Difference in means (t-test) • Difference in distributions (c2 test)

3. Data Analysis • Lecture Outline • Processing and Visualizing Data • Why do we do this? • Processing the Luria-Delbruck Data • Processing the Public Goods Data • Analyzing Data (using Excel) • Difference in means (t-test) • Difference in distributions (c2 test)

4. Massaging? Dressing-up? Focusing 3 P colonies 4 R colonies Handling Data • After a laboratory experiment or time out in the field, you will have several data points. • How should one process this (potentially voluminous) data? • Organize it (spreadsheet programs, like Excel, can help) • Process it • Investigate portions of the data set • Look at relevant descriptive statistics • Transform data points in a well-defined way • Combine data points in a well-defined way • Visualize it • Subject it to an appropriate statistical test *

5. Picture = Words  1000 • We are visual animals and often can see patterns when data is presented visually • Examples: • Pie-chart illustrates the distribution of values of a single variable • X-Y plot illustrates the form of the relationship between two variables • Paired histograms illustrate the relationship between the distributions of two variables. • The most appropriate picture will often depend on the data: • Categorical or quantitative? • Frequencies, counts or measurements? • Relationship between data points?

6. Data Analysis • Lecture Outline • Processing and Visualizing Data • Why do we do this? • Processing the Luria-Delbruck Data • Processing the Public Goods Data • Analyzing Data (using Excel) • Difference in means (t-test) • Difference in distributions (c2 test)

7. The Data • Go to our class website: • http://depts.washington.edu/kerrpost/Bio481/HomePage • On the DATA link, download the following Excel (xls) files: • “E3_LD_Processed_Data” • “E3_PG_Processed_Data” • Take care as you process and visualize the class data– the product of your efforts can be used directly in your first two lab reports.

8. Processing the Luria-Delbruck Data DAY 1: Tuesday DAY 3: Thursday COUNT ×24 48 hours at 37C COUNT 48 hours at 37C ×3 • We’ll start by computing some useful statistics: • Mean number of colonies on a rifampicin plate. • Variance in number of colonies on a rifampicin plate. • Total number of rifampicin plates (number of replicates in the class). • Next we will compile the full distribution of rifampicin plate counts: • Actual distribution (COUNTIF function will be useful) • Expected distribution (get ready to write a complicated function!) • Let’s plot these distributions. • Finally, let’s compute the density of cells in the original wells.

9. Data Analysis • Lecture Outline • Processing and Visualizing Data • Why do we do this? • Processing the Luria-Delbruck Data • Processing the Public Goods Data • Analyzing Data (using Excel) • Difference in means (t-test) • Difference in distributions (c2 test)

10. Processing the Public Goods Data DAY 1: Tuesday DAY 2: Wednesday DAY 3:Thursday COUNT BK26+ pBR BK26+pBR alone BK26+pBR alone 24 hours at 37C 24 hours at 37C 24 hours at 37C 24 hours at 37C 24 hours at 37C 24 hours at 37C 24 hours at 37C 24 hours at 37C 24 hours at 37C 24 hours at 37C COUNT BK27 alone BK27 alone BK27 monocultures ↑ competitions ↓ monocultures ↑ competitions ↓ COUNT BK26+ pBR COUNT Liquid comp. Liquid comp. Init. Mix amp COUNT COUNT BK27 Agar comp. Agar comp. • We’ll start by computing the densities when alone. • Next, let’s compute the relative fitnesses (BK26+pBR relative to BK27) & plot these.

11. Save Your Work • Save your work by renaming the data files: • “E3_LD_Processed_Data_YOUR_INITIALS” • “E3_PG_Processed_Data_YOUR_INITIALS” • Save these files on a thumb drive or email them to yourself. • We will continue to work on these during class today.

12. Data Analysis • Lecture Outline • Processing and Visualizing Data • Why do we do this? • Processing the Luria-Delbruck Data • Processing the Public Goods Data • Analyzing Data (using Excel) • Difference in means (t-test) • Difference in distributions (c2 test)

13. Number of flips Number of “heads” 4 1 12 3 100 25 Control High-protein Control High-protein High-protein Control frequency ♀e♂e How do we statistically evaluate data? • When you were a child, your father tells you he will let you stay up late if the result of a coin he flips is heads. • Suppose the coin comes up heads 25% of the time • Is your Dad using a fair coin? How would you evaluate this? • Suppose your hear that a high-protein diet during puberty leads to an increased height as an adult. • The mean height in a high protein treatment was 5’11” and the mean height in a control treatment was 5’5” • What would you feed your kids? How do you gauge this? • The New York Times has just done an expose about sexism in graduate admissions in a famous department of mathematics • While the number of male and female applicants was equal, the number of males admitted was greater. • Should an formal inquiry take place? How do you evaluate the data? ♀a♂a

14. The Data • Go to your email or thumb drive and download your processed data files: • “E3_LD_Processed_Data_YOUR_INITIALS” • “E3_PG_Processed_Data_YOUR_INITIALS” • Take care as you analyze the class data– the product of your efforts can be used directly in your first two lab reports.

15. William Sealy Gossett Student’s t-test • Using a pseudonym, “Student,” Gossett described a test for distinguishing the difference between means of 2 data sets. • The t-test uses the statistics from two groups of data (means and s.d.) to generate a third statistic (the t statistic). • If the two groups of data come from populations with the same mean, the t statistic has a characteristic distribution itself (note the shape will depend on the sample sizes). • If the computed t is extreme, then the chance that the two groups have equal means is slim (quantified by the p-value of the test). The means are significantly different if p<0.05. • Assumptions • Each datum is independent • Data is normally distributed • DEMO: Performing a t-test • Computing a p-value from a t-test • Distinguish the different types of t-tests: • Paired versus Unpaired data • Equal versus Unequal variance • One-tailed versus Two-tailed tests • How can we use t-tests for the Public Goods Data? What type of t-test is appropriate? How do you report it?

16. Data Analysis • Lecture Outline • Processing and Visualizing Data • Why do we do this? • Processing the Luria-Delbruck Data • Processing the Public Goods Data • Analyzing Data (using Excel) • Difference in means (t-test) • Difference in distributions (c2 test)

17. Karl Pearson c2 Test • Karl Pearson introduced a test to distinguish whether an observed set of frequencies differs from a specified frequency distribution. • The c2-test uses frequency data to generate a statistic (c2). • If the observed frequencies come from a population with the specified frequency, the c2 statistic has a characteristic distribution (the shape will depend on the # of classes). • If the computed c2 is extreme, then the chance that the observed frequencies derive from the specified distribution is slim (this is quantified by the p-value from the test). The observed frequencies are significantly different if p<0.05. • Assumptions • Frequencies are derived from independent sampling • There are not several frequencies that are very small • DEMO: Performing a chi-square test • Computing a p-value from a chi-square-test • Perform a c2 test to see if the Luria-Delbruck Data from the class differs from the frequencies expected under directed mutation. What can you conclude?