One-way Analysis of Variance

One-way Analysis of Variance

(ANOVA)

Note: In this version, we’ll use Excel to do all the calculations. If you wanted the tutorial that worked through the calculations using only a calculator, go back to the menu of tutorials to select that one.

Introduction, slide 1Once upon a time….

(that time was Fall 2012)

At a community college far, far away…

(actually, right here at Saddleback College, but “far, far away” sounds better)

Introduction, slide 2An evil biology professor…

That’s me!

Introduction, slide 3…enslaved an entire class in a room full of fancy equipment and made them do work! This was called….

Dr. Huntley’s Bio 3b Lab

Question: do light conditions affect chlorophyll concentration?The students were trying to determine whether light

conditions had a significant effect on the concentration of chlorophyll in geranium leaves. For their experiment, they had three groups of plants:

Acknowledgement: This image was found on Google images.It originated at http://geranamine.org/

Group 1: the light group

The first group was exposed to normal light conditions.

Aaah….I love sunshine!

Group 2: the dark group

The second group was kept in the dark for 72 hours.

Yikes! I’ve always been afraid of the dark!

Group 3: the 4-hour group

The third group was kept in the dark for 72 hours and then exposed to light for 4 hours.

Option to read about the experimentThen the students measured the chlorophyll concentration in the leaves. Click on Dr. Huntley if you’d like to read about how they did this.

The dataThere were 12 lab groups, each of which measured chlorophyll concentration in leaves from one plant from each group. The class then pooled their data; the results are in the table below. All data is given in .

LIGHT DARK DARK + 4 hr. Light56.02 25.97 46.7363.41 44.05 50.6760.35 51.31 45.3364.30 44.95 55.6458.95 38.58 47.3669.01 34.12 43.4269.52 37.94 51.9556.02 45.58 48.8955.26 27.25 45.2056.53 44.44 56.0258.07 49.02 54.1161.05 44.95 52.67

DirectionsPerform an ANOVA test with α = .05 to determine whether there is a significant difference in the mean concentration of chlorophyll between the 3 groups.

You may assume that chlorophyll concentration is approximately normally distributed.

If you want to work through this problem on your own and just check your work when you’re done, go ahead. Click on the leaf to the right when you’re ready to check your answer.

Set-upOpen a new Excel workbook and enter in the data.

Your worksheet should look like this.

Select “Data” from menu

From the top menu, select “Data.”

Select “Data Analysis” from menu

When this menu opens up, select “Data Analysis” from the far right.

What to do if “Data Analysis” option isn’t there

If “Data Analysis” doesn’t appear here, yell at your computer in frustration! Then install the Analysis Toolpak (you should have directions for doing so in one of your labs.)

The dialogue box for “Data Analysis”The dialogue box to the left will open up.

If “Anova: Single Factor” isn’t the first option, move this bar up until you see it.

Selecting the Anova option in Excel

The dialogue box to the right will open up.

Click on “Anova: Single Factor” to highlight it and then click on “OK.”

Filling in the input range

The above dialogue box will appear.

In the “Input Range” box, type in the cells that contain the headings (light, dark, and 4-hour) and the data, starting with the cell in the upper left-hand corner, followed by a colon (:), followed by the cell in the lower right-hand corner.

Finding the correct input rangeIn the “Input Range” box, type in the cells that contain the headings (light, dark, and 4-hour) and the data, starting with the cell in the upper left-hand corner, followed by a colon (:), followed by the cell in the lower right-hand corner.

From the screen shot below, we see that for this example that is A1:C13.

A1:C13

Alternate way to enter input range

From the screen shot below, we see that for this example that is A1:C13.

A1:C13 Note that you can also enter the input range by using the red arrow and highlighting the appropriate cells; feel free to use this option if you are familiar with it.

Select Grouped by “Columns” in dialogue box

A1:C13

Make sure you’ve checked the bubble that says the data is sorted by columns.

A1:C13

Click on this box to indicate that the first row contains the headings (labels) for our columns; the actual data starts in row 2. When you click on the box, a check mark will appear.

A1:C13

Click on this box to indicate that the first row contains the headings (labels) for our columns; the actual data starts in row 2. When you click on the box, a check mark will appear.

This is a neat feature of Excel; when we type in the headings and check this box, we’ll see these headings in the output generated by Excel. If we didn’t do this, the data would be described as “Column 1, Column 2,” and “Column 3” instead of “Light, Dark,” and “4-hour.”

Entering alpha value into dialogue box

A1:C13

Be sure the correct value of alpha appears in this box. If it doesn’t, click on the box and type in the correct value---for this problem, .05.

.05

Selecting “Output Range” from dialogue box

A1:C13

.05

Under “Output options,” select “Output Range” by clicking on the bubble next to it .

Entering the output range

A1:C13

.05

Pick an empty cell and enter it here. This is where your results will appear. I’ve chosen to enter E3. Again, if you prefer to use the red arrow and highlight the desired cell, that’s fine.

E3

Click “OK” when dialogue box is completed

A1:C13

.05

E3

Finally, click “OK.”

You’ll see the output boxes below, positioned to start in the cell you selected . Notice that it’s a little hard to read, since not all the information fits in the space allotted for it.

Adjusting the output box to make it easier to read

To fix this, click on “Home” at the top of your screen.

Adjusting the output box to make it easier to read, slide 2

When this menu opens up, click on “Format” in the options above “Cells.”

Adjusting the output box to make it easier to read, slide 3

Then select “Autofit Column Width” from the drop-down menu that opens up.

The output box from Excel, adjusted so it’s easier to read

Now we can read everything in the output box.

Ready to start hypothesis testNow we’re ready to perform our hypothesis test!

We’ll work through the usual 6-step process, and refer to the output from Excel whenever we need a calculated value.

Step 1: State the hypotheses and identify the claim (if there is one)

The Null hypothesis is always that there is no difference between the means. We can describe this in symbols.

𝐻𝑜 :𝜇𝑙=𝜇𝑑=𝜇4

Remember to use μ, the symbol for population mean. Notice that I’m using a subscript of “l” for the light group, “d” for the dark group, and “4” for the 4-hour group.

The alternate hypothesis𝐻𝑜 :𝜇𝑙=𝜇𝑑=𝜇4

The Alternate hypothesis is that there is a difference in the means. This includes the possibility

a) that we have 3 different means, or b) that 2 means are the same and only 1 is different from

the others. To allow for both possibilities, we state the Alternate hypothesis in words.

At least one mean is different from the others.

Note that this problem was posed as a question so that there is no claim.

Step (*)

Draw the F distribution and mark off the area in the right tail.

Slow Down!

There are 3 things we have to check first!

Checking to see if we can use the F-distribution1. Chlorophyll concentration is

approximately normally distributed. We were told to assume this.

2. We have independent samples. We can trust Dr. Huntley to set up a valid experiment.

3. The population variances are equal. This is a

tough one, since we don’t know any of the population variances.

Rule of thumb for checking that variances are equal

We’ll rely on the rule of thumb that it’s ok to proceed if the largest sample standard deviation (not the variance!) is no more than twice the smallest.

Excel gives us the sample variances. To calculate the sample standard deviations, we need to take the square roots.

Rule of thumb for checking that variances are equal, slide 2

We’ll rely on the rule of thumb that it’s ok to proceed if the largest sample standard deviation (not the variance!) is no more than twice the smallest.

LARGEST

smallest

The largest is less than twice the smallest.

The F-distributionSo we can go ahead and draw the F distribution and mark off the area in the right tail.

.05This test is always right-tailed (like me!) since if the means are different, that will make the F-value BIG.

Step 2

Mark off the critical value.

.05

The critical value is the boundary of the right tail; it goes here.

Find the critical value in the Excel output here.

Then add the critical value to our picture.

.05

3.284…

Step 3

Mark off the test value, F.

Finding the test value, F, on Excel’s output

Find the test value in the Excel output here.

Then add the test value to our picture.

.05

3.284… 33.411…

Since 33.411… > 3.284…, it goes to the right.

Step 4: Decide whether or not to reject the Null.

.05

3.284… 33.411…

The test value is in the critical region; reject the Null.

Step 5: Answer the question in plain English

There is enough evidence to support a claim that light conditions affect chlorophyll concentration.

Equivalently, there is enough evidence to reject a claim that light conditions don’t affect chlorophyll concentration.

Request for a summaryI’m exhausted! Could we just see a quick summary of all that?

Summary: review of Excel output

First we entered the data into Excel, selected the “Anova: Single Factor” option, and got the above output. Then we used this output to perform the usual 6 steps of hypothesis testing.

Summary: 6 steps of hypothesis testingEach click will give you one step; step (*) is broken into two clicks.

Step 1 At least one mean is different from the others.

Step(*)

.05

3.284…Step 2 33.411… Step 3

Step 4 Reject the Null.

Step 5 There is enough evidence to support a claim that light conditions affect chlorophyll concentration.

And there was much rejoicing.

Press escape to exit the slide show

If you keep clicking through the slides, you’ll go to the explanation of the experiment.

Steps 1 and 2 of the experiment

Step 1. Cuttwo 5 mm disksfrom a geranium leaf.

Acknowledgement: The image of the geranium leaf was found on Google images. It originated at davesgarden.com

Step 2. Soak disks in 5 ml of 8% acetone solution for 48 hours at

5 ml8 % acetone

solution

Step 3 of experimentStep 3. Use a spectrophotometer to read the absorbance. From this, calculate the chlorophyll concentration in micrograms of chlorophyll per milliliter of solution .

Acknowledgement: This image was found on google images, which credits it with coming from flushgordon.info

Step 4 of experimentStep 4. Convert from to . This is the data we’ll test.

For this tutorial, Dr. Huntley has kindly converted the data to the appropriate units for us.

Thanks Dr. Huntley!!

Please don’t mention it.Especially not to my 3b students; I make them do the conversions themselves.

Return to the main slide show after learning about the experimentClick on the picture below to see the data the students

collected.

Top SecretLab Results