STATISTIEK VOOR
PSYCHOLOGEN: DEEL 4
Lijst met formules
I. The good old one-way ANOVA
I.1. The ANOVA model: Notation and interpretation
𝑦𝑖𝑗
Score van een persoon i in conditie j
𝑚𝑒𝑡 𝑖 = 1, … , 𝑚𝑗 𝑒𝑛 𝑗 = 1, … , 𝑎
𝑎𝑙𝑙𝑒 𝑚𝑗 𝑧𝑖𝑗𝑛 𝑔𝑒𝑙𝑖𝑗𝑘
Gebalanceerd design
𝑎𝑙𝑙𝑒 𝑚𝑗 𝑧𝑖𝑗𝑛 𝑜𝑛𝑔𝑒𝑙𝑖𝑗𝑘
Ongebalanceerd design
Levels van een a factor a condities
𝑎
Totaal aantal participanten n = ∑ 𝑚j
𝑗=1
𝑚𝑗
∑ 𝑦ij
Sample average in condition j 𝑖=1
𝑦̅𝑗 =
𝑚𝑗
𝑚
𝑗 𝑚 𝑗
∑𝑎𝑗=1 ∑𝑖=1 𝑦𝑖𝑗 ∑𝑎𝑗=1 ∑𝑖=1 𝑦𝑖𝑗
Grand sample average 𝑦̅ = 𝑎 =
𝑛
∑ 𝑚j
𝑗=1
1
,I.2. Step 1: Models and hypotheses
𝑦𝑖𝑗 = 𝜇𝑗 + 𝜖𝑖𝑗′
Full model
𝑚𝑒𝑡 𝜖𝑖𝑗 ~𝑖𝑖𝑑 𝑁(0, 𝜎 2 )
Reduced model 𝑦𝑖𝑗 = 𝜇 + 𝜖𝑖𝑗′
(speciaal geval van full model) 𝑚𝑒𝑡 𝜖𝑖𝑗 ~𝑖𝑖𝑑 𝑁(0, 𝜎 2 )
𝑎 𝑚𝑗
Sum of squared differences 𝑄𝑅𝑒𝑑𝑢𝑐𝑒𝑑 (𝜇) = ∑ ∑(𝑦𝑖𝑗 − 𝜇)2
𝑗=1 𝑖=1
𝑎 𝑚𝑗
𝑄𝐹𝑢𝑙𝑙 (𝜇1 , … , 𝜇𝑎 ) = ∑ ∑(𝑦𝑖𝑗 − 𝜇𝑗 )2
𝑗=1 𝑖=1
Least squares function (full model)
Estimator for 𝜇𝑗 in the full model is equal to:
̂𝑓𝑢𝑙𝑙
𝜇̂𝑗 = 𝑦𝑖𝑗 = 𝑦̅𝑗
𝑆𝑆𝐸𝑅𝑒𝑑𝑢𝑐𝑒𝑑 = 𝑄𝑅𝑒𝑑𝑢𝑐𝑒𝑑 (𝜇̂ )
𝑎 𝑚𝑗
Error sum of squares
(summary measure of the size of the residuals) = ∑ ∑(𝑦𝑖𝑗 − 𝑦̅)2
Reduced 𝑗=1 𝑖=1
𝑆𝑆𝐸𝐹𝑢𝑙𝑙 = 𝑄𝐹𝑢𝑙𝑙 (𝜇
̂,
1 …,𝜇
̂)
𝑎
𝑎 𝑚𝑗
Full = ∑ ∑(𝑦𝑖𝑗 − 𝑦̅𝑗 )2
𝑗=1 𝑖=1
By imposing the H0 (en dus van full model
naar reduced model gaan), zal de data minder 𝑆𝑆𝐸𝑅𝑒𝑑𝑢𝑐𝑒𝑑 ≥ 𝑆𝑆𝐸𝐹𝑢𝑙𝑙
goed uitgelegd worden (of even goed)
𝑎 𝑚𝑗
Total sum of squares
(measures the total variation present in the 𝑆𝑆𝑇𝑜𝑡 = ∑ ∑(𝑦𝑖𝑗 − 𝑦̅)2
data) 𝑗=1 𝑖=1
2
, → bij een one-way ANOVA heb je de
speciale situatie:
𝑆𝑆𝑇𝑜𝑡 = 𝑆𝑆𝐸𝑅𝑒𝑑𝑢𝑐𝑒𝑑
𝑆𝑆𝐸𝑓𝑓 = 𝑆𝑆𝐸𝑅𝑒𝑑𝑢𝑐𝑒𝑑 − 𝑆𝑆𝐸𝐹𝑢𝑙𝑙′
→ bij one-way ANOVA:
Effect sum of squares
𝑎
𝑆𝑆𝐸𝑓𝑓 = ∑ 𝑚𝑗 ( 𝑦̅𝑗 − 𝑦̅)2
𝑗=1
𝑎
Degrees of freedom 𝑑𝑓𝑅𝑒𝑑𝑢𝑐𝑒𝑑 = ∑ 𝑚𝑗 − 1 = 𝑛 − 1
Reduced model 𝑗=1
𝑎
Full model 𝑑𝑓𝐹𝑢𝑙𝑙 = ∑ 𝑚𝑗 − 𝑎 = 𝑛 − 𝑎
𝑗=1
Simpler model have a larger degrees of
𝑑𝑓𝑅𝑒𝑑𝑢𝑐𝑒𝑑 > 𝑑𝑓𝐹𝑢𝑙𝑙
freedom
𝑆𝑆𝐸𝑅𝑒𝑑𝑢𝑐𝑒𝑑
𝑀𝑆𝐸𝑅𝑒𝑑𝑢𝑐𝑒𝑑 =
𝑛−1
𝑆𝑆𝐸𝐹𝑢𝑙𝑙
Mean squares 𝑀𝑆𝐸𝐹𝑢𝑙𝑙 =
𝑛−𝑎
𝑆𝑆𝐸𝑓𝑓
𝑀𝑆𝐸𝑓𝑓 =
𝑎−1
𝑎 𝑎 𝑎
Effect parameter of condition j = 𝛼𝑗
∑ 𝛼𝑗 = ∑(𝜇𝑗 − 𝜇) = ∑ 𝜇𝑗 − 𝑎𝜇 = 𝑎𝜇
(expresses the effect or deviation of condition
𝑗=1 𝑗=1 𝑗=1
j compared to the grand mean 𝜇) − 𝑎𝜇 = 0
3
PSYCHOLOGEN: DEEL 4
Lijst met formules
I. The good old one-way ANOVA
I.1. The ANOVA model: Notation and interpretation
𝑦𝑖𝑗
Score van een persoon i in conditie j
𝑚𝑒𝑡 𝑖 = 1, … , 𝑚𝑗 𝑒𝑛 𝑗 = 1, … , 𝑎
𝑎𝑙𝑙𝑒 𝑚𝑗 𝑧𝑖𝑗𝑛 𝑔𝑒𝑙𝑖𝑗𝑘
Gebalanceerd design
𝑎𝑙𝑙𝑒 𝑚𝑗 𝑧𝑖𝑗𝑛 𝑜𝑛𝑔𝑒𝑙𝑖𝑗𝑘
Ongebalanceerd design
Levels van een a factor a condities
𝑎
Totaal aantal participanten n = ∑ 𝑚j
𝑗=1
𝑚𝑗
∑ 𝑦ij
Sample average in condition j 𝑖=1
𝑦̅𝑗 =
𝑚𝑗
𝑚
𝑗 𝑚 𝑗
∑𝑎𝑗=1 ∑𝑖=1 𝑦𝑖𝑗 ∑𝑎𝑗=1 ∑𝑖=1 𝑦𝑖𝑗
Grand sample average 𝑦̅ = 𝑎 =
𝑛
∑ 𝑚j
𝑗=1
1
,I.2. Step 1: Models and hypotheses
𝑦𝑖𝑗 = 𝜇𝑗 + 𝜖𝑖𝑗′
Full model
𝑚𝑒𝑡 𝜖𝑖𝑗 ~𝑖𝑖𝑑 𝑁(0, 𝜎 2 )
Reduced model 𝑦𝑖𝑗 = 𝜇 + 𝜖𝑖𝑗′
(speciaal geval van full model) 𝑚𝑒𝑡 𝜖𝑖𝑗 ~𝑖𝑖𝑑 𝑁(0, 𝜎 2 )
𝑎 𝑚𝑗
Sum of squared differences 𝑄𝑅𝑒𝑑𝑢𝑐𝑒𝑑 (𝜇) = ∑ ∑(𝑦𝑖𝑗 − 𝜇)2
𝑗=1 𝑖=1
𝑎 𝑚𝑗
𝑄𝐹𝑢𝑙𝑙 (𝜇1 , … , 𝜇𝑎 ) = ∑ ∑(𝑦𝑖𝑗 − 𝜇𝑗 )2
𝑗=1 𝑖=1
Least squares function (full model)
Estimator for 𝜇𝑗 in the full model is equal to:
̂𝑓𝑢𝑙𝑙
𝜇̂𝑗 = 𝑦𝑖𝑗 = 𝑦̅𝑗
𝑆𝑆𝐸𝑅𝑒𝑑𝑢𝑐𝑒𝑑 = 𝑄𝑅𝑒𝑑𝑢𝑐𝑒𝑑 (𝜇̂ )
𝑎 𝑚𝑗
Error sum of squares
(summary measure of the size of the residuals) = ∑ ∑(𝑦𝑖𝑗 − 𝑦̅)2
Reduced 𝑗=1 𝑖=1
𝑆𝑆𝐸𝐹𝑢𝑙𝑙 = 𝑄𝐹𝑢𝑙𝑙 (𝜇
̂,
1 …,𝜇
̂)
𝑎
𝑎 𝑚𝑗
Full = ∑ ∑(𝑦𝑖𝑗 − 𝑦̅𝑗 )2
𝑗=1 𝑖=1
By imposing the H0 (en dus van full model
naar reduced model gaan), zal de data minder 𝑆𝑆𝐸𝑅𝑒𝑑𝑢𝑐𝑒𝑑 ≥ 𝑆𝑆𝐸𝐹𝑢𝑙𝑙
goed uitgelegd worden (of even goed)
𝑎 𝑚𝑗
Total sum of squares
(measures the total variation present in the 𝑆𝑆𝑇𝑜𝑡 = ∑ ∑(𝑦𝑖𝑗 − 𝑦̅)2
data) 𝑗=1 𝑖=1
2
, → bij een one-way ANOVA heb je de
speciale situatie:
𝑆𝑆𝑇𝑜𝑡 = 𝑆𝑆𝐸𝑅𝑒𝑑𝑢𝑐𝑒𝑑
𝑆𝑆𝐸𝑓𝑓 = 𝑆𝑆𝐸𝑅𝑒𝑑𝑢𝑐𝑒𝑑 − 𝑆𝑆𝐸𝐹𝑢𝑙𝑙′
→ bij one-way ANOVA:
Effect sum of squares
𝑎
𝑆𝑆𝐸𝑓𝑓 = ∑ 𝑚𝑗 ( 𝑦̅𝑗 − 𝑦̅)2
𝑗=1
𝑎
Degrees of freedom 𝑑𝑓𝑅𝑒𝑑𝑢𝑐𝑒𝑑 = ∑ 𝑚𝑗 − 1 = 𝑛 − 1
Reduced model 𝑗=1
𝑎
Full model 𝑑𝑓𝐹𝑢𝑙𝑙 = ∑ 𝑚𝑗 − 𝑎 = 𝑛 − 𝑎
𝑗=1
Simpler model have a larger degrees of
𝑑𝑓𝑅𝑒𝑑𝑢𝑐𝑒𝑑 > 𝑑𝑓𝐹𝑢𝑙𝑙
freedom
𝑆𝑆𝐸𝑅𝑒𝑑𝑢𝑐𝑒𝑑
𝑀𝑆𝐸𝑅𝑒𝑑𝑢𝑐𝑒𝑑 =
𝑛−1
𝑆𝑆𝐸𝐹𝑢𝑙𝑙
Mean squares 𝑀𝑆𝐸𝐹𝑢𝑙𝑙 =
𝑛−𝑎
𝑆𝑆𝐸𝑓𝑓
𝑀𝑆𝐸𝑓𝑓 =
𝑎−1
𝑎 𝑎 𝑎
Effect parameter of condition j = 𝛼𝑗
∑ 𝛼𝑗 = ∑(𝜇𝑗 − 𝜇) = ∑ 𝜇𝑗 − 𝑎𝜇 = 𝑎𝜇
(expresses the effect or deviation of condition
𝑗=1 𝑗=1 𝑗=1
j compared to the grand mean 𝜇) − 𝑎𝜇 = 0
3
