Course recap
Applied Statistics for Life Sciences
Example: body temperatures
Is the true mean body temperature actually 98.6°F?
The data provide evidence that mean body temperature differs from 98.6°F (T = -2.24 on 129 degrees of freedom, p = 0.027). With 95% confidence, the mean is estimated to be between 98.12 and 98.57°F.
Example: hours of sleep
Do U.S. adults sleep an average of 8 hours per night? (NHANES data)
The data provide evidence that the average U.S. adult does not sleep 8 hours per night (T = -42.53 on 3178 degrees of freedom, p < 0.0001). With 95% confidence, the mean nightly hours of sleep among U.S. adults is estimated to be between 6.91 and 7.01 hours.
Example: desired weight
Does the average U.S. adult wish to lose weight?
One Sample t-test
data: weight.diffs
t = 4.2172, df = 59, p-value = 4.311e-05
alternative hypothesis: true mean is greater than 0
95 percent confidence interval:
10.99824 Inf
sample estimates:
mean of x
18.21667 The data provide evidence that the average U.S. adult’s actual weight exceeds their desired weight (T = 4.2172 on 59 degrees of freedom, p < 0.0001).
Example: finch beak depth
Did beak depth increase after the 1977 drought on Daphne Major?
Welch Two Sample t-test
data: depth by year
t = -4.5833, df = 172.98, p-value = 4.37e-06
alternative hypothesis: true difference in means between group 1976 and group 1978 is less than 0
95 percent confidence interval:
-Inf -0.427321
sample estimates:
mean in group 1976 mean in group 1978
9.469663 10.138202 The data provide evidence that mean beak depth increased following the drought (T = -4.57 on 111.79 degrees of freedom, p < 0.0001).
Example: diet and longevity
Does caloric restriction affect mean lifespan in mice?
Df Sum Sq Mean Sq F value Pr(>F)
diet 3 11426 3809 87.41 <2e-16 ***
Residuals 233 10152 44
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Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1The data provide evidence that diet restriction has an effect on mean lifetime among mice (F = 87.41 on 3 and 233 degrees of freedom, p < 0.0001). With 95% confidence, the proportion of variation in lifespan explained by diet is estimated to be between 44% and 60%.
Example: sign test
Does the median DDT concentration in kale exceed the 3ppm safety threshold?
The data provide evidence that median DDT in kale exceeds 3ppm (p = 0.019).
Example: rank sum test
Is there a difference in serum cholesterol between corn flake and oat bran diets?
The data provide no evidence of a difference in cholesterol between diets (rank sum test, p = 0.74).
Example: Kruskal-Wallis test
Does the distribution of AAM lengths differ by geographic location?
The data provide evidence that the distribution of AAM lengths differs by geographic location (Kruskal-Wallis test, p = 0.0025).
Example: cognitive function and age
How much does RFFT score decline with age? (PREVEND data)
Call:
lm(formula = rfft ~ age, data = prevend)
Residuals:
Min 1Q Median 3Q Max
-56.085 -14.690 -2.937 12.744 77.975
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 134.0981 6.0701 22.09 <2e-16 ***
age -1.1908 0.1007 -11.82 <2e-16 ***
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Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 20.52 on 206 degrees of freedom
Multiple R-squared: 0.4043, Adjusted R-squared: 0.4014
F-statistic: 139.8 on 1 and 206 DF, p-value: < 2.2e-16With each additional year of age, mean RFFT score decreases by an estimated 1.19 points.
Example: diabetes prevalence
Is diabetes prevalence in the U.S. equal to 10%?
1-sample proportions test with continuity correction
data: 57 out of 500, null probability 0.1
X-squared = 0.93889, df = 1, p-value = 0.3326
alternative hypothesis: true p is not equal to 0.1
95 percent confidence interval:
0.08814952 0.14594579
sample estimates:
p
0.114 The data provide no evidence that diabetes prevalence differs from 10% (\(\chi^2\) = 0.94 on 1 degree of freedom, p = 0.33). With 95% confidence, diabetes prevalence among U.S. adults is estimated to be between 8.8% and 14.6%.
Extensions: multiple regression
What if we have more than one explanatory variable?
\[\text{% change} = \beta_0 + \beta_1\, \text{age} + \beta_2\; \text{sex} + \epsilon\]
(Intercept) age sexMale
91.86 -1.20 -22.80 Each line represents the relationship between age and strength gain after adjusting for sex.
- Females gain ~22% more strength than males of the same age
- Each additional year of age is associated with ~1% less strength gain
Extensions: two-way ANOVA
What if we have two grouping variables — and they might interact?
Df Sum Sq Mean Sq F value Pr(>F)
genotype 2 7043 3522 3.713 0.0250 *
sex 1 80847 80847 85.246 <2e-16 ***
genotype:sex 2 5837 2918 3.077 0.0468 *
Residuals 589 558609 948
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Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1- Interaction: does the genotype effect differ by sex?
- Parallel lines → no interaction
- Non-parallel lines → interaction present