bot3015l data analysis and interpretation presentation created by jean burns and sarah tso all...

BOT3015LData analysis and

interpretation

Presentation created by Jean Burns and Sarah Tso

All photos from Raven et al. Biology of Plants except when otherwise noted

Today

•Types of data

•Discrete, Continuous

•Independent, dependent

•Types of statistics

•Descriptive, Inferential

•Creating graphs in excel

•Doing a t-test or Chi Square

•Lab: create graphs and do statistics for the gas exchange experiment

Today

•Types of data

•Discrete, Continuous

•Independent, dependent

•Types of statistics

•Descriptive, Inferential

•Creating graphs in excel

•Doing a t-test

•Lab: create graphs and do statistics for the gas exchange experiment

Types of data

1. Discrete: Having categories (i.e. flowers present/flowers absent, large/medium/small)

Seed heteromorphism: a discrete character.

Not hetermorphicHetermorphic

Types of data

1. Discrete: Having categories (i.e. flowers present/flowers absent, large/medium/small)

2. Continuous: Having infinite possible values (i.e. age, growth rate)

Seed size: a continuous character

Commelina benghalensis seed size variation

Types of data

1. Independent: Manipulated or selected with the hypothesis that it is causally linked to the dependent variable. Cause.

2. Dependent: Measured as a response to the independent variable. Effect.

Independent and dependent variables

Independent: Treatment (CO2 concentration)

Dependent: Number of open and closed stomata, or stomatal aperture

Assumption: Changes in CO2 concentration will affect stomatal aperture.

Today

•Types of data

•Discrete, Continuous

•Independent, dependent

•Types of statistics

•Descriptive, Inferential

•Creating graphs in excel

•Doing a t-test

•Lab: create graphs and do statistics for the gas exchange experiment

Types of statistics

1. Descriptive: Summarize a set of data.

2. Inferential: Draw conclusions from a data set.

Types of statistics

1. Descriptive: Summarize a set of data.

2. Inferential: Draw conclusions from a data set.

Mean: a type of descriptive statistic

Arithmetic mean

http://www.steve.gb.com/science/statistics.html

Mean: a type of descriptive statisticMeasure of the central tendency of a data set.

Fre

quen

cy

Value

Mean = 2.9

Standard deviation: a type of descriptive statistic

Standard deviation

http://www.steve.gb.com/science/statistics.html

Standard deviation: a type of descriptive statistic.

Measure of spread of variability in a data set.

Fre

quen

cy

Value

Standard deviation = 0.25

Standard deviation: a type of descriptive statistic.

Measure of spread of variability in a data set.

Fre

quen

cy

Value

Standard deviation = 0.58 Standard deviation = 0.41

Value

Types of statistics

1. Descriptive: Summarize a set of data.

2. Inferential: Draw conclusions from a data set.

Pearson’s 2: a type of inferential statistic

Used on discrete response variable, when you have discrete treatments (independent variables).

Example: The number of open and closed stomata in response to lower CO2 concentration.

t-test: a type of inferential statistic

Used on continuous response variable, when you have discrete treatments (independent variables).

Example: Stomatal aperture response to lower CO2 concentration.

Regression: a type of inferential statistic

Used on continuous response variable, when you have continuous treatments (independent variables).

Example: Stomatal aperture response to varied CO2 concentration (when the CO2 concentration is actually measured).

*Talk to your TA if you want to know how to do this

Observation: both internal and external factors affect stomatal aperture

Question: What is the effect of CO2 concentration on stomatal aperture or the number of open and closed stomata?

Experimental Design

Question: What is the effect of reducing CO2 concentration on the number of open stomata?

Treatment: Reduce CO2 concentration using sodium hydroxide:

CO2 + NaOH => NaHCO3 (sodium bicarbonate)

Control: Ambient atmospheric CO2 concentration

Data: Count the number of open and closed stomata (are these data discrete or continuous?)

Hypothesis testing for discrete data

Pearson’s Chi Square (2): a test of association between to categorical variables.

Ho: Both treatments yield an equal number of open and closed stomata.HA1: NaOH treatment results in fewer open stomata than the control.HA2: NaOH treatment results in more open stomata than the control.

Step 1: Make a contingency table

# open stomata

# closed stomata

NaOH 5 15

Ambient CO2 15 5

This is a 2 x 2 contingency table, having two columns and two rows, but it can have other dimensions.

Step 2: Make a contingency table

# open stomata

# closed stomata

Row Totals

NaOH 5 15 20

Ambient CO2 15 5 20

Column Totals 20 20 N = 40

Add the row and column totals and the grand total, N.

Step 3: Calculate expected values based on null hypothesis

# open stomata

# closed stomata

Row Totals

NaOH 5 (10) 15 (10) 20

Ambient CO2 15 (10) 5 (10) 20

Column Totals 20 20 N = 40

Ho: Both treatments yield an equal number of open and closed stomata.For each cell, the expected value is:Row total x column total/ N.

Step 4: Calculate the 2 and degrees of freedom

2 = {(observed - expected)2/ expected}

d.f. = (# of columns - 1) x (# of rows - 1)

# open stomata

# closed stomata

Row Totals

NaOH 5 (10) 15 (10) 20

Ambient CO2 15 (10) 5 (10) 20

Column Totals 20 20 N = 40

2 = (5 - 10)2/ 10 + (15 - 10)2/10 + (15 - 10)2/10 + (5 - 10)2/ 10 = 10

d.f. = (2 - 1) x (2 - 1) = 1

Step 4: Compare calculated 2 with the critical value from a Chi Square distribution table

The critical value can be obtained from a table based on the degrees of freedom and the level of confidence, which is set at P = 0.05.

2 calc = 10

2 crit = 3.84, d.f. = 1

If the calculated value exceeds the critical value, you can reject your Ho

Hypothesis testing for continuous data

Ho: Both treatments yield the same

stomatal aperture.

HA1: NaOH treatment results in narrower stomatal aperture.

HA2: NaOH treatment results in larger

stomatal aperture.

Hypothesis testing for continuous data

Ho: Both treatments yield the same

stomatal aperture.

HA1: Water treatment results in larger

stomatal aperture.

HA2: NaOH treatment results in larger

stomatal aperture.

A t-test will distinguish

between Ho and HA, then you

must look at the direction of the difference to interpret the

results.

We will use a t-test for this example:

http://www.steve.gb.com/science/statistics.html

Question: is there a difference in the means between two treatments?

Large overlap = not different.http://www.steve.gb.com/science/statistics.html

Question: is there a difference in the means between two treatments?

Large overlap = not different.http://www.steve.gb.com/science/statistics.html

small

larget < ~2

Question: is there a difference in the means between two treatments?

Large overlap = not different.http://www.steve.gb.com/science/statistics.html

Question: is there a difference in the means between two treatments?

Little overlap = different.http://www.steve.gb.com/science/statistics.html

larger

larget > ~2

Question: is there a difference in the means between two treatments?

Little overlap = different.http://www.steve.gb.com/science/statistics.html

Question: is there a difference in the means between two treatments?

Little overlap = different.http://www.steve.gb.com/science/statistics.html

large

smallt > ~2

What if the answer is not so obvious?

This is why we need statistics.

Degrees of freedom

• DF = n1 + n2 - 2

DF = number of independent categories in a statistical test.

For example, in a t-test, we are estimating 2 parameters the mean and the variance. Thus we subtract 2 from the degrees of freedom, because 2 elements are no longer independent.

DF is a measure of a test’s power. Larger sample sizes (and DF) result in more power to detect differences between the means.

t-value distribution

http://www.psychstat.missouristate.edu/introbook/sbk25m.htm

t-value

freq

uenc

y

1. Get tcrit from a table of t-values, for P = 0.05 and the correct DF.2. If tobserved > tcrit, then the test is significant.3. If P < 0.05, the means are different.

Factors influencing a difference between means

• Distance between means

• Variance in each sample (Standard Deviation, SD)

• T-value (means and SD)

• Number of samples (DF)

• Level of error we are willing to accept to consider two means different (P-value).

Today

•Types of data

•Discrete, Continuous

•Independent, dependent

•Types of statistics

•Descriptive, Inferential

•Creating graphs in excel

•Doing a t-test

•Lab: create graphs and do statistics for the gas exchange experiment

Creating graphs in excel1. Open excel (Start/Applications/Microsoft Excel)2. Enter the data in table format

Creating graphs in excel1. Open excel (Start/Applications/Microsoft Excel)2. Enter the data in table format3. In the cells directly under treatment data:

Creating graphs in excel1. Open excel (Start/Applications/Microsoft Excel)2. Enter the data in table format3. Calculate the mean and standard deviation

Mean: enter formula

=average(cells to calculate the mean from)

Example:

=AVERAGE(A2:A11)

Creating graphs in excel1. Open excel (Start/Applications/Microsoft Excel)2. Enter the data in table format3. Calculate the mean and standard deviation

Standard deviation: enter formula

=stdev(cells to calculate the mean from)

Example:

=STDEV(A2:A11)

Creating graphs in excel1. Open excel (Start/Applications/Microsoft Excel)2. Enter the data in table format3. Calculate the mean and standard deviation4. Select the data you wish to graph

Select these cells

Creating graphs in excel1. Open excel (Start/Applications/Microsoft Excel)2. Enter the data in table format3. Calculate the mean and standard deviation4. Select the data you wish to graph5. Click the chart button or “Insert” “Chart…”

Chart Button

Creating graphs in excel1. Open excel (Start/Applications/Microsoft Excel)2. Enter the data in table format3. Calculate the mean and standard deviation4. Select the data you wish to graph5. Click the chart button6. Chose your chart options:

• Column (next)• Series/Category x-axis labels/highlight

treatment labels (next)• Titles/label axes including Units (next)• Finish

Now your chart should look like this:

Creating graphs in excel1. Open excel (Start/Applications/Microsoft Excel)2. Enter the data in table format3. Calculate the mean and standard deviation4. Select the data you wish to graph5. Click the chart button6. Chose your chart options7. Add error bars to your chart:

• Double click on the bar• Y-error bars (at the top)• Go to Custom• Select the cells with the standard deviation*Note: you should only have error bars if the data

are continuous.

Now your chart should look like this:

Today

•Types of data

•Discrete, Continuous

•Independent, dependent

•Types of statistics

•Descriptive, Inferential

•Creating graphs in excel

•Doing a t-test

•Lab: create graphs and do statistics for the gas exchange experiment

Performing a t-test

In this course, we will demonstrate the use of Excel for statistics; however, more advanced software, designed specifically for statistical analyses, offer more detailed analyses. Use the software of your choice, being sure to indicate the software that is used.

t-test with Excel

In excel:1. In an empty cell, “Insert” a “Function”2. Find “T-TEST”3. “Array 1” is one set of values. Include each value (e.g.

each aperture size under one condition)4. “Array 2 is the other set of values (e.g. each aperture

size under the other condition.5. We will be performing a two-tailed distribution t-test.

Enter “2” in “tails.”6. We are assuming there is equal variance for the two

samples, so enter “2” in “type.”7. “OK” will return the probability (p) value. This is the

probability that the difference between the sets of values is random.

Reminders

Report submissions (paper and turnitin)refer to “organization of a lab report” in the beginning of your lab manual. • Titles must be descriptive• Methods must be complete• Results should include descriptions (in your own

words) not just graphs and tables (although those are also necessary).

• Discussion must demonstrate thought• Submit copies of your references with your reports