Statistics for EES 3. From standard error to t-test
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1 Statistics for EES 3. From standard error to t-test Dirk Metzler May 23, 2011
2 Contents 1 The Normal Distribution 2 Taking standard errors into account 3 The t-test for paired samples Example: Orientation of pied flycatchers
3 Contents The Normal Distribution 1 The Normal Distribution 2 Taking standard errors into account 3 The t-test for paired samples Example: Orientation of pied flycatchers
4 The Normal Distribution Density of the normal distribution Normaldichte µ + σ µ µ + σ
5 The Normal Distribution Density of the normal distribution Normaldichte µ + σ µ µ + σ The normal distribution is also called Gauß distribution
6 The Normal Distribution Density of the normal distribution The normal distribution is also called Gauß distribution (after Carl Friedrich Gauß, )
7 The Normal Distribution dbinom(400:600, 1000, 0.5) :600
8 The Normal Distribution dbinom(400:600, 1000, 0.5) :600
9 The Normal Distribution Density of standard normal distribution A random variable Z with density f (x) = 1 2π e x2 2 Gaussian bell is called standard normally distributed.
10 The Normal Distribution Density of standard normal distribution A random variable Z with density f (x) = 1 2π e x2 2 Gaussian bell for short: Z N (0, 1) is called standard normally distributed.
11 The Normal Distribution Density of standard normal distribution A random variable Z with density f (x) = 1 2π e x2 2 Gaussian bell for short: Z N (0, 1) EZ = 0 Var Z = is called standard normally distributed.
12 The Normal Distribution If Z is N (0, 1) distributed, then X = σ Z + µ is normally distributed with mean µ and variance σ 2, for short: X N (0, 1)
13 The Normal Distribution If Z is N (0, 1) distributed, then X = σ Z + µ is normally distributed with mean µ and variance σ 2, for short: X N (0, 1) X has the density f (x) = 1 2πσ e (x µ)2 2σ 2.
14 The Normal Distribution Keep in mind If Z N (µ, σ 2 ), then: Pr( Z µ > σ) < 1 3
15 The Normal Distribution Keep in mind If Z N (µ, σ 2 ), then: Pr( Z µ > σ) < 1 3 Pr( Z µ > 2 σ) < 0.05
16 The Normal Distribution Keep in mind If Z N (µ, σ 2 ), then: Pr( Z µ > σ) < 1 3 Pr( Z µ > 2 σ) < 0.05 Pr( Z µ > 3 σ) < 0.003
17 The Normal Distribution f (x) = 1 e (x µ)2 2σ 2 2πσ µ σ µ µ + σ
18 The Normal Distribution Densities need integrals If Z is a random variable with density f (x) then a b, Pr(Z [a, b]) =
19 The Normal Distribution Densities need integrals If Z is a random variable with density f (x) then Pr(Z [a, b]) = a b, b a f (x)dx.
20 The Normal Distribution Example: Let Z N (µ = 5, σ 2 = 2.25). Pr(Z < a) can be computed with pnorm(a,mean=5,sd=1.5)
21 The Normal Distribution Example: Let Z N (µ = 5, σ 2 = 2.25). Pr(Z < a) can be computed with pnorm(a,mean=5,sd=1.5) Computation of Pr(Z [3, 4]):
22 The Normal Distribution Example: Let Z N (µ = 5, σ 2 = 2.25). Pr(Z < a) can be computed with pnorm(a,mean=5,sd=1.5) Computation of Pr(Z [3, 4]): Pr(Z [3, 4]) = Pr(Z < 4) Pr(Z < 3)
23 The Normal Distribution Example: Let Z N (µ = 5, σ 2 = 2.25). Pr(Z < a) can be computed with pnorm(a,mean=5,sd=1.5) Computation of Pr(Z [3, 4]): Pr(Z [3, 4]) = Pr(Z < 4) Pr(Z < 3) > pnorm(4,mean=5,sd=1.5)-pnorm(3,mean=5,sd=1.5)
24 The Normal Distribution Example: Let Z N (µ = 5, σ 2 = 2.25). Pr(Z < a) can be computed with pnorm(a,mean=5,sd=1.5) Computation of Pr(Z [3, 4]): Pr(Z [3, 4]) = Pr(Z < 4) Pr(Z < 3) > pnorm(4,mean=5,sd=1.5)-pnorm(3,mean=5,sd=1.5) [1]
25 The Normal Distribution Let Z N (µ, σ 2 ). How to compute Pr(Z = 5)?
26 The Normal Distribution Let Z N (µ, σ 2 ). How to compute Pr(Z = 5)? Answer: For each x R holds Pr(Z = x) = 0
27 The Normal Distribution Let Z N (µ, σ 2 ). How to compute Pr(Z = 5)? Answer: For each x R holds Pr(Z = x) = 0 How about EZ = a a Pr(Z = a)?
28 The Normal Distribution Let Z N (µ, σ 2 ). How to compute Pr(Z = 5)? Answer: For each x R holds Pr(Z = x) = 0 How about EZ = a a Pr(Z = a)? Needs a new definition: EZ = x f (x)dx
29 The Normal Distribution Let Z N (µ, σ 2 ). How to compute Pr(Z = 5)? Answer: For each x R holds Pr(Z = x) = 0 How about EZ = a a Pr(Z = a)? Needs a new definition: EZ = x f (x)dx But we already know the result EZ = µ.
30 Contents Taking standard errors into account 1 The Normal Distribution 2 Taking standard errors into account 3 The t-test for paired samples Example: Orientation of pied flycatchers
31 Taking standard errors into account Important consequence Consider the interval x s/ n x + s/ n x
32 Taking standard errors into account Important consequence This interval contains µ with probability of ca. 2/3 x s/ n x + s/ n x
33 Taking standard errors into account Important consequence This interval contains µ with probability of ca. 2/3 x s/ n x + s/ n x probability that µ is outside of interval is ca. 1/3
34 Taking standard errors into account Thus: It may happen that x deviates from µ by more than s/ n.
35 Taking standard errors into account Application 1: Which values of µ are plausible?
36 Taking standard errors into account Application 1: Which values of µ are plausible? x = 0.12 s/ n = 0.007
37 Taking standard errors into account Application 1: Which values of µ are plausible? x = 0.12 s/ n = Question: Could it be that µ = 0.115?
38 Taking standard errors into account Answer: Yes, not unlikely.
39 Taking standard errors into account Answer: Yes, not unlikely. Deviation x µ = =
40 Taking standard errors into account Answer: Yes, not unlikely. Deviation x µ = = Standard Error s/ n = 0.007
41 Taking standard errors into account Answer: Yes, not unlikely. Deviation x µ = = Standard Error s/ n = Deviations like this are not unusual.
42 Taking standard errors into account Answer: Yes, not unlikely. Deviation x µ = = Standard Error s/ n = Deviations like this are not unusual.
43 Taking standard errors into account Application 2: Comparison of mean values
44 Taking standard errors into account Application 2: Comparison of mean values Example: Galathea
45 Taking standard errors into account Galathea: Carapace lengths in a sample Males: x 1 = 3.04 mm s 1 = 0.9 mm n 1 = 25 Females: x 2 = 3.23 mm s 2 = 0.9 mm n 2 = 29
46 Taking standard errors into account The females are apparently larger.
47 Taking standard errors into account The females are apparently larger. Is this significant?
48 Taking standard errors into account The females are apparently larger. Is this significant? Or could it be just random?
49 Taking standard errors into account How precisely do we know the true mean value?
50 Taking standard errors into account How precisely do we know the true mean value? Males: x 1 = 3.04 mm s 1 = 0.9 mm n 1 = 25 s 1 / n 1 = 0.18 [mm]
51 Taking standard errors into account How precisely do we know the true mean value? Males: x 1 = 3.04 mm s 1 = 0.9 mm n 1 = 25 s 1 / n 1 = 0.18 [mm] We have to assume uncertainty in the magnitude of ±0.18 (mm) in x 1
52 Taking standard errors into account How precisely do we know the true mean value?
53 Taking standard errors into account How precisely do we know the true mean value? Females: x 2 = 3.23 mm s 2 = 0.9 mm n 2 = 29 s 2 / n 2 = 0.17 [mm]
54 Taking standard errors into account How precisely do we know the true mean value? Females: x 2 = 3.23 mm s 2 = 0.9 mm n 2 = 29 s 2 / n 2 = 0.17 [mm] It is not unlikely that x 2 deviates from the true mean by more than ±0.17 (mm).
55 Taking standard errors into account The difference of the means = 0.19 [mm]
56 Taking standard errors into account The difference of the means = 0.19 [mm] is not much larger than the expected inaccuracies.
57 Taking standard errors into account The difference of the means = 0.19 [mm] is not much larger than the expected inaccuracies. It could also be due to pure random that x 2 > x 1
58 Taking standard errors into account MORE PRECISELY: If the true means are actually equal µ Females = µ Males
59 Taking standard errors into account MORE PRECISELY: If the true means are actually equal µ Females = µ Males it is still quite likely that the sample means x 2 are x 1 that different.
60 Taking standard errors into account In the language of statistics: The difference of the mean values is (statistically) not significant.
61 Taking standard errors into account not significant = can be just random
62 Taking standard errors into account Application 3: If the mean values are represented graphically, you should als show their precision (±s/ n)
63 Taking standard errors into account Not Like this: Carapace lengths: Mean values for males and females Carapace length [mm] Males Females
64 Taking standard errors into account Like that: Carapace lengths: Mean values ± standard errors for males and females Carapace length [mm] Males Females
65 Taking standard errors into account Application 4: Planning an experiment:
66 Taking standard errors into account Application 4: Planning an experiment: How many observations do I need? (How large should n be?)
67 Taking standard errors into account If you know which precision you need for (the standard error s/ n of) x
68 Taking standard errors into account If you know which precision you need for (the standard error s/ n of) x and if you already have an idea of s
69 Taking standard errors into account If you know which precision you need for (the standard error s/ n of) x and if you already have an idea of s then you can estimate the value of n that is necessary: s/ n = g (g = desired standard error)
70 Taking standard errors into account Example: Stressed transpiration values in another sorghum subspecies: x = 0.18 s = 0.06 n = 13
71 Taking standard errors into account Example: Stressed transpiration values in another sorghum subspecies: x = 0.18 s = 0.06 n = 13 How often do we have to repeat the experiment to get a standard error of 0.01?
72 Taking standard errors into account Example: Stressed transpiration values in another sorghum subspecies: x = 0.18 s = 0.06 n = 13 How often do we have to repeat the experiment to get a standard error of 0.01? Which n do we need?
73 Taking standard errors into account Solution: desired: s/ n 0.01
74 Taking standard errors into account Solution: desired: s/ n 0.01 From the previous experiment we know: s 0.06,
75 Taking standard errors into account Solution: desired: s/ n 0.01 From the previous experiment we know: s 0.06, so: n 6
76 Taking standard errors into account Solution: desired: s/ n 0.01 From the previous experiment we know: s 0.06, so: n
77 Summary Taking standard errors into account Assume a population has mean value µ and standard deviation σ.
78 Taking standard errors into account Summary Assume a population has mean value µ and standard deviation σ. We draw a sample of size n from this population with sample mean x.
79 Taking standard errors into account Summary Assume a population has mean value µ and standard deviation σ. We draw a sample of size n from this population with sample mean x. x is a random variable
80 Taking standard errors into account Summary Assume a population has mean value µ and standard deviation σ. We draw a sample of size n from this population with sample mean x. x is a random variable with mean value µ and standard deviation σ/ n.
81 Taking standard errors into account Summary Assume a population has mean value µ and standard deviation σ. We draw a sample of size n from this population with sample mean x. x is a random variable with mean value µ and standard deviation σ/ n. Estimate the standard deviation of x by s/ n.
82 Taking standard errors into account Summary Assume a population has mean value µ and standard deviation σ. We draw a sample of size n from this population with sample mean x. x is a random variable with mean value µ and standard deviation σ/ n. Estimate the standard deviation of x by s/ n. s/ n is the Standard Error (of the Mean).
83 Taking standard errors into account Summary Assume a population has mean value µ and standard deviation σ. We draw a sample of size n from this population with sample mean x. x is a random variable with mean value µ and standard deviation σ/ n. Estimate the standard deviation of x by s/ n. s/ n is the Standard Error (of the Mean). Deviations of x of the magnitude of s/ n are usual.
84 Taking standard errors into account Summary Assume a population has mean value µ and standard deviation σ. We draw a sample of size n from this population with sample mean x. x is a random variable with mean value µ and standard deviation σ/ n. Estimate the standard deviation of x by s/ n. s/ n is the Standard Error (of the Mean). Deviations of x of the magnitude of s/ n are usual. They are not significant: they can be random.
85 Contents The t-test for paired samples 1 The Normal Distribution 2 Taking standard errors into account 3 The t-test for paired samples Example: Orientation of pied flycatchers
86 The t-test for paired samples sample means ± standard errors A B C D Is A significantly different from B?
87 The t-test for paired samples sample means ± standard errors A B C D Is A significantly different from B? No: A is in 2/3 range of µ B
88 The t-test for paired samples sample means ± standard errors A B C D Is B significantly different from C?
89 The t-test for paired samples sample means ± standard errors A B C D Is B significantly different from C? No: 2/3 ranges of µ B and µ C overlap
90 The t-test for paired samples sample means ± standard errors A B C D Is C significantly different from D?
91 The t-test for paired samples sample means ± standard errors A B C D Is C significantly different from D? We do not know yet.
92 The t-test for paired samples sample means ± standard errors A B C D Is C significantly different from D? We do not know yet. (Answer will be no.)
93 Contents The t-test for paired samples Example: Orientation of pied flycatchers 1 The Normal Distribution 2 Taking standard errors into account 3 The t-test for paired samples Example: Orientation of pied flycatchers
94 The t-test for paired samples Example: Orientation of pied flycatchers Pied flycatchers (Ficedula hypoleuca) Foto (c) Simon Eugster hypoleuca NRM.jpg
95 The t-test for paired samples Example: Orientation of pied flycatchers Wiltschko, W.; Gesson, M.; Stapput, K.; Wiltschko, R. Light-dependent magnetoreception in birds: interaction of at least two different receptors. Naturwissenschaften 91.3, pp , Wiltschko, R.; Ritz, T.; Stapput, K.; Thalau, P.; Wiltschko, W. Two different types of light-dependent responses to magnetic fields in birds. Curr Biol 15.16, pp , Wiltschko, R.; Stapput, K.; Bischof, H. J.; Wiltschko, W. Light-dependent magnetoreception in birds: increasing intensity of monochromatic light changes the nature of the response. Front Zool, 4, 2007.
96 The t-test for paired samples Example: Orientation of pied flycatchers
97 The t-test for paired samples Example: Orientation of pied flycatchers Direction of flight with blue light.
98 The t-test for paired samples Example: Orientation of pied flycatchers Direction of flight with blue light. Another flight of the same bird with blue light.
99 The t-test for paired samples Example: Orientation of pied flycatchers All flight s directions of this bird in blue light.
100 The t-test for paired samples Example: Orientation of pied flycatchers All flight s directions of this bird in blue light.
101 The t-test for paired samples Example: Orientation of pied flycatchers All flight s directions of this bird in blue light. corresponding exit points
102 The t-test for paired samples Example: Orientation of pied flycatchers Directions of this bird s flights in green light.
103 The t-test for paired samples Example: Orientation of pied flycatchers Directions of this bird s flights in green light. corresponding exit points
104 The t-test for paired samples Example: Orientation of pied flycatchers corresponding exit points arrow heads: barycenter of exit points for green light
105 The t-test for paired samples Example: Orientation of pied flycatchers arrow heads: barycenter of exit points for green light
106 The t-test for paired samples Example: Orientation of pied flycatchers arrow heads: barycenter of exit points for green light the same for blue directions
107 The t-test for paired samples Example: Orientation of pied flycatchers arrow heads: barycenter of exit points for green light the same for blue directions the more variable the directions the shorter the arrows!
108 The t-test for paired samples Question of interest Example: Orientation of pied flycatchers Does the color of monochromatic light influence the orientation?
109 The t-test for paired samples Question of interest Example: Orientation of pied flycatchers Does the color of monochromatic light influence the orientation? Experiment: For 17 birds compare barycenter vector lengths for blue and green light.
110 The t-test for paired samples Example: Orientation of pied flycatchers Pies flycathchers : barycenter vector lengths for green light and for blue light b, n=17 with green light with blue light
111 The t-test for paired samples Example: Orientation of pied flycatchers Compute for each bird the distance to the diagonal,
112 The t-test for paired samples Example: Orientation of pied flycatchers Compute for each bird the distance to the diagonal, i.e. x := green length blue length
113 The t-test for paired samples Example: Orientation of pied flycatchers Compute for each bird the distance to the diagonal, i.e. x := green length blue length
114 The t-test for paired samples Example: Orientation of pied flycatchers Can the theoretical mean be µ = 0? (i.e. the expectation value for a randomly chosen bird)
115 The t-test for paired samples Example: Orientation of pied flycatchers Can the theoretical mean be µ = 0? (i.e. the expectation value for a randomly chosen bird) x =
116 The t-test for paired samples Example: Orientation of pied flycatchers Can the theoretical mean be µ = 0? (i.e. the expectation value for a randomly chosen bird) x = s =
117 The t-test for paired samples Example: Orientation of pied flycatchers Can the theoretical mean be µ = 0? (i.e. the expectation value for a randomly chosen bird) x = s = SEM = s n
118 The t-test for paired samples Example: Orientation of pied flycatchers Can the theoretical mean be µ = 0? (i.e. the expectation value for a randomly chosen bird) x = s = SEM = s = n 17
119 The t-test for paired samples Example: Orientation of pied flycatchers Can the theoretical mean be µ = 0? (i.e. the expectation value for a randomly chosen bird) x = s = SEM = s = n 17 = 0.022
120 The t-test for paired samples Example: Orientation of pied flycatchers Can the theoretical mean be µ = 0? (i.e. the expectation value for a randomly chosen bird) x = s = SEM = s = n 17 = 0.022
121 The t-test for paired samples Example: Orientation of pied flycatchers Is x µ a large deviation?
122 The t-test for paired samples Example: Orientation of pied flycatchers Is x µ a large deviation? Large?
123 The t-test for paired samples Example: Orientation of pied flycatchers Is x µ a large deviation? Large? Large compared to what?
124 The t-test for paired samples Example: Orientation of pied flycatchers Is x µ a large deviation? Large? Large compared to what? Large compared the the standard error?
125 The t-test for paired samples Example: Orientation of pied flycatchers x µ s/ n
126 The t-test for paired samples Example: Orientation of pied flycatchers x µ s/ n So, x is more than 2.3 standard deviations away from µ = 0.
127 The t-test for paired samples Example: Orientation of pied flycatchers x µ s/ n So, x is more than 2.3 standard deviations away from µ = 0. How probable is this when 0 is the true theoretical mean?
128 The t-test for paired samples Example: Orientation of pied flycatchers x µ s/ n So, x is more than 2.3 standard deviations away from µ = 0. How probable is this when 0 is the true theoretical mean? in other words: Is this deviation significant?
129 The t-test for paired samples Example: Orientation of pied flycatchers We know: x µ σ/ n is asymptotically (for large n) standard-normally distributed.
130 The t-test for paired samples Example: Orientation of pied flycatchers We know: x µ σ/ n is asymptotically (for large n) standard-normally distributed. σ is the theoretical standard deviation, but we do not know it.
131 The t-test for paired samples Example: Orientation of pied flycatchers We know: x µ σ/ n is asymptotically (for large n) standard-normally distributed. σ is the theoretical standard deviation, but we do not know it. Instead of σ we use the sample standard deviation s.
132 The t-test for paired samples Example: Orientation of pied flycatchers We know: x µ σ/ n is asymptotically (for large n) standard-normally distributed. σ is the theoretical standard deviation, but we do not know it. Instead of σ we use the sample standard deviation s. We have applied the 2/3 rule of thumb which is based on the normal distribution.
133 The t-test for paired samples Example: Orientation of pied flycatchers We know: x µ σ/ n is asymptotically (for large n) standard-normally distributed. σ is the theoretical standard deviation, but we do not know it. Instead of σ we use the sample standard deviation s. We have applied the 2/3 rule of thumb which is based on the normal distribution. For questions about the significance this approximation is too imprecise if σ is replaced by s (and n is not extremely large).
134 The t-test for paired samples Example: Orientation of pied flycatchers General Rule If X 1,..., X n are independently drawn from a normal distributed with mean µ and then s 2 = 1 n 1 n (X i X) 2, i=1 X µ s/ n is t-distributed with n 1 degrees of freedom (df).
135 The t-test for paired samples Example: Orientation of pied flycatchers General Rule If X 1,..., X n are independently drawn from a normal distributed with mean µ and then s 2 = 1 n 1 n (X i X) 2, i=1 X µ s/ n is t-distributed with n 1 degrees of freedom (df). The t-distribution is also called Student-distribution, since Gosset published it using this synonym.
136 The t-test for paired samples Density of the t-distribution Example: Orientation of pied flycatchers density Density of t-distribution with 16 df
137 The t-test for paired samples Density of the t-distribution Example: Orientation of pied flycatchers density Density of t-distribution with 16 df Density of standard normal distribution
138 The t-test for paired samples Density of the t-distribution Example: Orientation of pied flycatchers density Density of t-distribution with 16 df Density of standard normal distribution µ = 0 and σ 2 = 16 14
139 The t-test for paired samples Density of the t-distribution Example: Orientation of pied flycatchers density Density of t-distribution with 5 df Density of normal distribution with µ = 0 and σ 2 = 5 3
140 The t-test for paired samples Example: Orientation of pied flycatchers How (im)probable is a deviation by 2.35 standard errors? Pr(T = 2.34) =
141 The t-test for paired samples Example: Orientation of pied flycatchers How (im)probable is a deviation by 2.35 standard errors? Pr(T = 2.34) = 0
142 The t-test for paired samples Example: Orientation of pied flycatchers How (im)probable is a deviation by at least 2.35 standard errors? Pr(T = 2.34) = 0 does not help!
143 The t-test for paired samples Example: Orientation of pied flycatchers How (im)probable is a deviation by at least 2.35 standard errors? Pr(T = 2.34) = 0 does not help! Compute Pr(T 2.34), the so-called p-value. density
144 The t-test for paired samples Example: Orientation of pied flycatchers How (im)probable is a deviation by at least 2.35 standard errors? Pr(T = 2.34) = 0 does not help! Compute Pr(T 2.34), the so-called p-value. density The total area of the magenta-colored regions is the p-value
145 The t-test for paired samples Example: Orientation of pied flycatchers R does that for us: > pt(-2.34,df=16)+pt(2.34,df=16,lower.tail=false) [1]
146 The t-test for paired samples Example: Orientation of pied flycatchers R does that for us: > pt(-2.34,df=16)+pt(2.34,df=16,lower.tail=false) [1] Comparison with normal distribution: > pnorm(-2.34)+pnorm(2.34,lower.tail=false) [1] > pnorm(-2.34,sd=16/14)+ + pnorm(2.34,sd=16/14,lower.tail=false) [1]
147 t-test with R The t-test for paired samples Example: Orientation of pied flycatchers > x <- length$green-length$blue > t.test(x) One Sample t-test data: x t = , df = 16, p-value = alternative hypothesis: true mean is not equal to 0 95 percent confidence interval: sample estimates: mean of x
148 The t-test for paired samples Example: Orientation of pied flycatchers We note: p Wert =
149 The t-test for paired samples Example: Orientation of pied flycatchers We note: p Wert = i.e.: If the Null hypothesis just random, in our case µ = 0 holds, any deviation like the observed one or even more is very improbable.
150 The t-test for paired samples Example: Orientation of pied flycatchers We note: p Wert = i.e.: If the Null hypothesis just random, in our case µ = 0 holds, any deviation like the observed one or even more is very improbable. If we always reject the null hypothesis if and only if the p-value is below a level of significance of 0.05, then we can say following holds:
151 The t-test for paired samples Example: Orientation of pied flycatchers We note: p Wert = i.e.: If the Null hypothesis just random, in our case µ = 0 holds, any deviation like the observed one or even more is very improbable. If we always reject the null hypothesis if and only if the p-value is below a level of significance of 0.05, then we can say following holds: If the null hypothesis holds, then the probability that we reject it, is only 0.05.
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