Biostatistics for Animal Science

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6 Biostatistics for Animal Science 1.4 Numerical Methods for Presenting Data Numerical methods for presenting data are often called descriptive statistics. They include: a) measures of central tendency; b) measures of variability ; c) measures of the shape of a distribution; and d) measures of relative standing. a) measures of central tendency b) measures of variability Descriptive statistics c) measures of the shape of a distribution d) measures of relative position - arithmetic mean - range - skewness - percentiles - median - variance - kurtosis - z-values - mode - standard deviation - coefficient of variation Before descriptive statistics are explained in detail, it is useful to explain a system of symbolic notation that is used not only in descriptive statistics, but in statistics in general. This includes the symbols for the sum, sum of squares and sum of products. 1.4.1 Symbolic Notation The Greek letter Σ (sigma) is used as a symbol for summation, and yi for the value for observation i. The sum of n numbers y1, y2,…, yn can be expressed: Σi yi = y1 + y2 +.....+ yn The sum of squares of n numbers y1, y2,…, yn is: Σi y 2 i = y 2 1 + y 2 2 +.....+ y 2 n The sum of products of two sets of n numbers (x1, x2,…, xn) and (y1, y2,…, yn): Σi xiyi = x1y1 + x2y2 +.....+ xnyn Example: Consider a set of three numbers: 1, 3 and 6. The numbers are symbolized by: y1 = 1, y2 = 3 and y3 = 6. The sum and sum of squares of those numbers are: Σi yi = 1 + 3 + 6 = 10
Σi y 2 i = 1 2 + 3 2 + 6 2 = 46 Consider another set of numbers: x1 = 2, x2 = 4 and x3 = 5. The sum of products of x and y is: Σi xiyi = (1)(2) + (3)(4) + (6)(5) = 44 Three main rules of addition are: Chapter 1 Presenting and Summarizing Data 7 1. The sum of addition of two sets of numbers is equal to the addition of the sums: Σi (xi + yi) = Σi xi + Σi yi 2. The sum of products of a constant k and a variable y is equal to the product of the constant and the sum of the values of the variable: Σi k yi = k Σi yi 3. The sum of n constants with value k is equal to the product n k: Σi k = n k 1.4.2 Measures of Central Tendency Commonly used measures of central tendency are the arithmetic mean, median and mode. The arithmetic mean of a sample of n numbers y1,y2,..., yn is: ∑ n y i i y = The arithmetic mean for grouped data is: ∑ f y i i i y = n with fi being the frequency or proportion of observations yi. If fi is a proportion then n = 1. Important properties of the arithmetic mean are: ∑ 1. ( − ) i yi y = 0 The sum of deviation from the arithmetic mean is equal to zero. This means that only (n - 1) observations are independent and the n th can be expressed as y ny − y − − y n = 1 ... n−1 ∑ 2. ( y − ) 2 minimum = i i y The sum of squared deviations from the arithmetic mean is smaller than the sum of squared deviations from any other value.
Page 1: Biostatistics for Animal Science
Page 4 and 5: CABI Publishing is a division of CA
Page 6 and 7: vi Biostatistics for Animal Science
Page 8 and 9: viii Biostatistics for Animal Scien
Page 10 and 11: x Biostatistics for Animal Science
Page 12 and 13: Preface This book was written to se
Page 15 and 16: Chapter 1 Presenting and Summarizin
Page 17 and 18: Brown 5% Simmental 76% Holstein 19%
Page 19: Number of calves 18 16 14 12 10 8 6
Page 25 and 26: 1.4.5 Measures of Relative Position
Page 29 and 30: Chapter 2 Probability The word prob
Page 31 and 32: 2.2.1 Multiplicative Rule Chapter 2
Page 33 and 34: Chapter 2 Probability 19 event. As
Page 35 and 36: Chapter 2 Probability 21 P(A ∩ B)
Page 37 and 38: 2.4 Bayes Theorem First calf Second
Page 39 and 40: We define: P(A1) = 0.6 is the proba
Page 41 and 42: Chapter 3 Random Variables and thei
Page 45 and 46: 3.2.3 Binomial Distribution Chapter
Page 47 and 48: ⎛10⎞ 0 10 0 10 b) P ( y = 0) =
Page 49 and 50: E(y) = µ = λ and Var(y) = σ 2 =
Page 53 and 54: 2.5% µ−1.96σ µ−σ µ µ+σ F
Page 61 and 62: e f ( y) = − 1 2 −1 ( y −µ )
Page 65 and 66: The expectation and variance of the
Page 67 and 68: Chapter 4 Population and Sample A p
Page 69 and 70: Chapter 4 Population and Sample 55
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5.2 Maximum Likelihood Estimation C
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Chapter 5 Estimation of Parameters
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By setting both terms to zero the e
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Chapter 5 Estimation of Parameters
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Chapter 6 Hypothesis Testing The fo
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µ 0 y − z ≈ s n Chapter 6 Hypo
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Known values are: y = 4000 kg σ =
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α -zα Figure 6.7 The critical val
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s( y 1− y2 ) = 2 s1 s + n n 1 2 2
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The estimated pooled variance is: s
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Chapter 6 Hypothesis Testing 77 tha
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( y − y2 ) − 0 ( 11. 857 − 21
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Normal Approximation Z 2.0626 One-S
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pˆ − pˆ 1 The estimator has var
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2 χ [ yi E( yi ) ] E( y ) ∑ −
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∑ ∑ y i i p0 = i = 1,..., k n i
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Chapter 6 Hypothesis Testing 89 Exp
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Chapter 6 Hypothesis Testing 91 An
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Decision of a statistical test H 0
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Chapter 6 Hypothesis Testing 95 nor
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Chapter 6 Hypothesis Testing 97 One
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0.45 0.40 0.35 0.30 0.25 0.20 0.15
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SAS output: alpha n mi0 mi1 stdev d
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6.12 Sample Size Chapter 6 Hypothes
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SAS program: DATA a; DO n = 2 TO 10
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Chapter 6 Hypothesis Testing 107 PR
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Chapter 7 Simple Linear Regression
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The measurements of the dependent v
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2 SSRES s = = MSRES = 115. 826 n
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Confidence intervals follow the cla
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spread yi about y A) y y * * * * *
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SS REG ( SSxy ) = SS xx 2 Chapter 7
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E(MSREG) = σ 2 + β1 2 SSxx Chapte
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The likelihood function when H0 is
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2 R = SS SS REG TOT Chapter 7 Simpl
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= (X'X) -1 X'y A maximum likelihood
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DATA a; alpha=0.05; n=6; b=7.52941;
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Chapter 8 Correlation 147 of correl
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has a t-distribution with (n - 2) d
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Chapter 8 Correlation 151 under H0:
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Exercises Chapter 8 Correlation 153
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The assumptions of the model are: 1
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where: Chapter 9 Multiple Linear Re
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The residual mean square, which is
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The null and alternative hypotheses
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Chapter 9 Multiple Linear Regressio
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yi = the weight of bull i x1i = the
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SAS program: DATA bulls; INPUT weig
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9.6.1 Analysis of Residuals Chapter
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y * * 1 * * * * * * * * * 2 * * * 4
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where: Chapter 9 Multiple Linear Re
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SAS program: DATA bull; INPUT weigh
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where: SSRES = residual mean square
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Chapter 10 Curvilinear Regression I
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Chapter 10 Curvilinear Regression 1
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The variance estimate is s 2 = 210.
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SAS program: DATA a; INPUT age weig
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Weight (g) 1000 800 600 400 200 0 0
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Chapter 11 One-way Analysis of Vari
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From the assumptions it follows: Ch
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By defining: SS SS SS TOT TRT RES =
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1) Total sum: Σi Σj yij = (270 +
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2 L( µ , σ | y) = The log likelih
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And as shown previously: Thus: 2
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The hypotheses can be tested using
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ANOVA table: Chapter 11 One-way Ana
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F 3 = 2890 2 296. 67 = 4. 871 Chapt
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SAS output: Chapter 11 One-way Anal
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The effect of sow 1 is: ˆ 1 τ = 0
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The log likelihood that is to be ma
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Solution for Random Effects Chapter
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11.3.1.2 Estimating Parameters Chap
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The sums of squares needed for test
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The Xr'Xr matrix and its inverse ar
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11.3.2 The Random Effects Model 11.
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11.3.2.4 Restricted Maximum Likelih
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Chapter 12 Concepts of Experimental
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12.5 Required Number of Replication
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SAS output: alfa df1 df2 n power 0.
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Chapter 13 Blocking 273 environment
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1) Total sum: Σi Σj yij 2) Correc
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Chapter 13 Blocking 277 Example: Th
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13.1.3 SAS Example for Block Design
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Animal No. (Treatment) Blocks I II
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Mean square for treatments: Mean sq
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= s y − y MS ij . i' j' . RES ⎛
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1347. 90 F = = 7. 67 175. 83 F test
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Dependent Variable: d_gain i/j 1 2
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Chapter 13 Blocking 291 the adjuste
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Exercises Chapter 13 Blocking 293 1
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µ = the overall mean τi = the fix
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BLOCK II The hypotheses are: Chapte
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The results are shown in the ANOVA
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Chapter 14 Change-over Designs 301
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3) Total (corrected) sum of squares
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3) Total (corrected) sum of squares
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Least Squares Means Adjustment for
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Chapter 15 Factorial Experiments A
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1) Total sum: Σi Σj Σk yijk 2) C
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Chapter 15 Factorial Experiments 31
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5) Sum of squares for vitamin II: (
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Chapter 15 Factorial Experiments 32
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Chapter 16 Hierarchical or Nested D
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6) Sum of squares within factor B (
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3) Total sum of squares: Chapter 16
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SAS output of the NESTED procedure:
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Chapter 17 More about Blocking If t
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Chapter 17 More about Blocking 333
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SAS program: DATA steer; INPUT pen
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SAS output of the GLM procedure: De
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Chapter 17 More about Blocking 339
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The ANOVA table is: Source Degrees
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The model for this design is: where
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Chapter 18 Split-plot Design 345 su
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Least Squares Means Chapter 18 Spli
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Source Degrees of freedom Factor A
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Effects Means Estimators Standard e
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Chapter 18 Split-plot Design 353 Ex
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Chapter 19 Analysis of Covariance A
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DATA gain; INPUT treatment $ initia
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Chapter 19 Analysis of Covariance 3
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For males (M) the model is: E(yi) =
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Chapter 19 Analysis of Covariance 3
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Chapter 20 Repeated Measures Experi
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Chapter 20 Repeated Measures 367 Ex
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8 9 1 1.3 8 10 1 1.3 8 11 1 1.3 8 1
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SAS output: Covariance Parameter Es
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where: Chapter 20 Repeated Measures
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Chapter 20 Repeated Measures 375 Ex
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σ ⎡ σ b σ ⎡ ⎤ 0 b0b1 [ 1 t
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⎡1 ⎢ ⎢ 1 R ˆ = ⎢1 ⎢ ⎣1
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⎡1 ⎢ ˆR ⎢ 1 = ⎢1 ⎢ ⎣1
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Chapter 20 Repeated Measures 383
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Chapter 21 Analysis of Numerical Tr
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21.2.1 SAS Example for Polynomial C
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Chapter 22 Discrete Dependent Varia
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where: pi = the proportion with mas
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Analysis Of Parameter Estimates Cha
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SAS program: DATA a; INPUT n y farm
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Farm The model is: where: Total no.
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LR Statistics For Type 1 Analysis C
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Solutions of Exercises 1.1. Mean =
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Appendix A: Vectors and Matrices A
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⎡a A + B = ⎢ ⎢ a ⎢⎣ a A +
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a = (X'X) -1 X'y Appendix A: Vector
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Critical Values of Student t Distri
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Critical Values of Chi-square Distr
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Critical Values of F Distributions,
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Critical Value of F Distributions,
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Critical Values of the Studentized
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References 437 Gianola, D. and Hamm
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Subject Index accuracy, 266 Akaike
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y hypothesis testing, 56 by paramet
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hypothesis test, 186 t test, 186 ra
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of regression estimators, 119, 159
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