Biomeasurement

Continued from last week:

t- and Mann-Whitney U tests

Background reading: Chapter 7

www.biomeasurement.net

Lecture Content• Comparing t- & MWU tests

• t-test

• Mann Whitney U test

Mann-Whitney U Test

• Comparison to t-Test

• When to use

• Example data

• Four steps

• Example from literature

Bone Density(g/cm2)

Females Males

0.972 0.9050.732 1.0160.874 0.8730.943 0.741.024 0.8610.755 0.8170.779 0.8971.007 0.9620.816 0.8510.755 0.8210.871 0.7630.721 0.8760.727 0.9440.796 0.9930.612 0.7740.775 0.7850.849 0.8920.773 1.0760.649 0.8880.865 0.865

Brine Shrimp Length (mm)

MediumSalinity

HighSalinity

5.5 6.06.0 7.05.0 7.57.0 6.05.5 7.56.0 8.07.0 11.08.0 9.06.0 8.08.0 11.06.0 8.07.0 8.06.0 7.07.0 7.06.0 7.08.0 9.06.0 .7.0 .7.5 .6.0 .7.5 .

• Difference

• Two samples

• Unrelated

SimilaritiesBirth weight (kg) of babies born to

Non smokers Heavy smokers

3.99 3.183.79 2.843.60 2.903.73 3.273.21 3.853.60 3.524.08 3.233.61 2.763.83 3.603.31 3.754.13 3.593.26 3.633.54 2.383.51 2.342.71

Differencest-test• Parametric• Scale data only

Mann-Whitney U test• Nonparametric• Scale or ordinal data

Speed of swimmingMedium Salinity High

Salinity

Slow FastFast SlowFast Very Fast

Medium FastSlow Medium

Very slow Very slowVery Fast SlowVery Slow Very Fast

Slow Very FastFast Very Fast

Medium FastSlow VerySlow

Time spentPlaying

(hours/day)

Enrichment Control

4.5 3.54.3 6.35.3 7.86.4 4.34.9 4.63.7 5.15.2

When to use• Difference.

• Two samples.

• Unrelated data.

• Dependent variable: Scale level

Do not use if...

• Comparing frequency distributions.

Bone Density(g/cm2)

Females Males

0.972 0.9050.732 1.0160.874 0.8730.943 0.741.024 0.8610.755 0.8170.779 0.8971.007 0.9620.816 0.8510.755 0.8210.871 0.7630.721 0.8760.727 0.9440.796 0.9930.612 0.7740.775 0.7850.849 0.8920.773 1.0760.649 0.8880.865 0.865

Example Data

Tests on counts/frequencies Tests of Relationship

Tests of Difference

1 set of Categories 2 sets of Categories

For scale/ordinal data from two variables

Regression Correlation

Scatterplot ScatterplotOnePie Chart

TwoPie Charts

Parametric Nonparametric

For counts/frequencies in categories

For scale/ordinal dependent variable data in categories distinguished by the independent variable

©Hawkins & Carter 2004

Pie Charts

Errorplots or Boxplot

Errorplots Boxplots

Choosing Chart

for Graphs

Scatterplots

The Four Steps

1. Construct a Null Hypothesis (Ho).

2. Decide Critical Significance Level ).

3. Calculate Statistic.

4. Reject or Accept the Null Hypothesis.

1. Construct Ho

Ho: There is no difference between the bone density of males and females over 50 years old.

In general no difference between the samples.

2. Decide

5% = 0.05.

3. Calculate Statistic.

U

U1 = n1 n 2 + n2(n2 + 1) - R2

2

U2 = n1 n 2 + n1(n1 + 1) - R1

2

Sum of ranks of sample 2

Size of sample 1

U is the the lower value of U1 or U2.

Check: U1 + U2 = n1 n 2

Bone Density Measurement (g/cm2)Females Males

0.972 35 0.905 310.732 5 1.016 380.874 26 0.873 250.943 32 0.740 61.024 39 0.861 210.755 7.5 0.817 170.779 13 0.897 301.007 37 0.962 340.816 16 0.851 200.755 7.5 0.821 180.871 24 0.763 90.721 3 0.876 270.727 4 0.944 330.796 15 0.993 360.612 1 0.774 110.775 12 0.785 140.849 19 0.892 290.773 10 1.076 400.649 2 0.888 280.865 22.5 0.865 22.5

330.5 489.5R2

R1

More about R

U1 = 20x20 + 20(20 + 1) – 489.5 2 =120.5

U2 = 20x20 + 20(20 + 1) – 330.5 2 =279.5

U

U = 120.5

n1 = 20

n2 = 20

Bone Density(g/cm2)

Females Males

0.972 0.9050.732 1.0160.874 0.8730.943 0.741.024 0.8610.755 0.8170.779 0.8971.007 0.9620.816 0.8510.755 0.8210.871 0.7630.721 0.8760.727 0.9440.796 0.9930.612 0.7740.775 0.7850.849 0.8920.773 1.0760.649 0.8880.865 0.865

Using SPSS

Dependent Variable

Independent Variable

4. Reject or Accept.

Using Critical Values.

Reject if your t is bigger than tcritical.

Using P Values

Reject if P is less than or equal to

Where do you get these from?

Using Critical Values.

120.5< 273 Reject

Bone density between males & females over 50 is different

From...

Step 2: = 0.05

Step 3: U=120.5, n1=20, n2=20

If U </= Ucritical REJECT H0 significant result.

n1

1 2 3 4 5 16 17 18 19 20n2 1 - - - - - - - - - -

2 - - - - - 31 32 34 36 383 - - - - 15 42 45 47 50 524 - - - 16 19 53 57 60 63 665 - - 15 19 23 65 68 72 76 80

16 - 31 42 53 65 181 191 202 212 22217 - 32 45 57 68 191 202 213 224 23518 - 34 47 60 72 202 213 225 236 24819 - 36 50 63 76 212 224 236 248 26120 - 38 52 66 80 222 235 248 261 273

Using P Values.

If P </= reject the null hypothesis.If P > accept the null hypothesis.If P </= reject the null hypothesis.If P > accept the null hypothesis.

0.032< 0.05 Reject

Bone density between males & females over 50 is different

Example from LiteratureDrews (1995), BEHAVIOUR 133The pattern and context of injuries was studied in a troop of yellow baboons in Mikumi National Park (Tanzania).

TABLE 7. Median and range of healing times in days for baboon injuries.

N Median Min MaxTotal sample* 35 23.0 8 120Small injuries (<5cm) 15 14.0 8 114Large injuries (>5cm) 11 24.0 10 120* The sample is composed of 24 cuts, 4 punctures, 1 tear, 1 case of

limping, 1 bruise, and 3 injuries of unspecified shape.

Quote from results: The difference between small and large wounds (Table 7) was not statistically significant (Mann-Whitney test, Z= -1.461, N1=15, N2=11, p=0.14).

TABLE 7. Median and range of healing times in days for baboon injuries.

N Median Min MaxTotal sample* 35 23.0 8 120Small injuries (<5cm) 15 14.0 8 114Large injuries (>5cm) 11 24.0 10 120* The sample is composed of 24 cuts, 4 punctures, 1 tear, 1 case of

limping, 1 bruise, and 3 injuries of unspecified shape.

Quote from results: The difference between small and large wounds (Table 7) was not statistically significant (Mann-Whitney test, Z= -1.461, N1=15, N2=11, p=0.14).

What is Z? - bone example revisted.

Back to baboons….QUESTIONS:• What were the two samples? Sample

sizes?

• What might the raw data have looked like?

• Why might he have used a Mann-Whitney U test instead of a t-test?

Lecture Content

• Comparing t- & MWU tests

• t-test

• Mann Whitney U test