Statistical test

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Statistical test (統計的検定)

Mount Usu / Sarobetsu post-mined peatland
From left: Crater basin in 1986 and 2006. Cottongrass / Daylily

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Terminology

zero-inflated model = hurdle model = two-part model

Types of statistical tests (検定の種類)

= parameter test + goodness-of-fit test

Parameter test

Goodness-of-fit test

Ex. Uniformity test (一様性の検定)
Sun 17. Mon 6. Tue 8. Wed 12. Thu 11. Fri 16. Sat 14 = Total 84

H: p_i = 1/7(I = 1 … 7) m_i = mp_i → 84 × 1/7 = 12 (> 5)
χ² = 1/12{(17 - 12)² + (6 - 12)² + (8 - 12)² + (12 - 12)²

+ (11 - 12)² + (16 - 12)² + (14 - 12)²} ≈ 82

χ²_0.05(df = n - 1 = 6) = 12.592 → D = [12.592, ∞],
accept H (= accept uniformity)

Softwares

SAS
SPSS (statistical packages for social sciences)
Systat
Statistica
MVSP (multi-variate statistical package)
JMP: good for biostatistics

R (package): freeware

1997 Ihaka R & Gentleman: proposed R → open source (⇔ S)
CRAN (The comprehensive R archive network): mega-infromation source on R

Python: freeware

Cpython

Analysis of variance (ANOVA, 分散分析)

an analysis tool used in statistics that splits an observed aggregate variability found inside a data set into two parts: systematic factors and random factors

systematic factors: a statistical influence on the given data set
random factors: … do not

One-way analysis of variance (one-way ANOVA, 1元配置分散分析)

used to determine whether there are any statistically significant differences between the means of three or more independent (unrelated) groups

Two-way analysis of variance (two-way ANOVA, 2元配置分散分析)

an extension of the one-way ANOVA that reveals the results of two independent variables on a dependent variable

[ ordination ]

Multivariate analysis (多変量解析)

Univariate analysis, which looks at just one variable
Bivariate analysis, which analyzes two variables
Multivariate analysis, which looks at more than two variables

[cluster analysis, meta-analysis, ordination]

Covariance structure analysis (共分散構造分析)

≈ structural equation model (構造方程式モデル) (with latent variable), SEM/LV
considering a model representing how various aspects of some phenomenon are thought to causally connect to one another

Structural equation modeling, SEM (構造方程式モデル)

primarily used in the social and behavioral sciences

now applied to various research

System dynamics, SD (システムダイナミクス)

Forrester, Jay Wright (1918-2016, MIT)
1944 started to develop a flight simulator, by a digital computer, Whirlwind
1956 develped system dynamics - to analyze social systems
1961 "Industrial Dynamics": business behavior simulation (early stage) ⇒
1969 "Urban Dynamics": diversified applications, e.g., urban planning
⇒ consolidation and integration = SD
1971 "World dynamics": affected The Limits to Growth (LTG, 成長の限界)

Advantages and applications

Diagrammatically describe causal relationships between model elements

→ automatic generation of numerical simulation models

convenient to understand relationships between elements

→ directly model individual problem phenomena and causal relationships

∴ suitable for simulation models for systems (society, business, policy, etc.) that are difficult to conduct experiment with or to oversee in a broad range

System (システム)

The concept of system for using system dynamics

system has the objective(s) (目的)
present broundary (システム境界) enclosing components

= the components within the system →

the components are connected logically to each other
input from the outsides (environments) →

[negative feedback] → output from the inside

System = natural system + human system

human system = social system* + physical system

*: present decision-making step (意思決定点) - real system

Optner, Stanford L (1920-2017)
Def. problem: defined as a situation in which there are two states: one is characterized by the present state, the other by a proposed state. The present state is exemplified by the existing system; the proposed state is exemplified by the system that is hypothesized (desired) or proposed (1965)
System analysis ≈ problem resolution

Factor analysis, FA (因子分析)

A part of general linear model
Assumpstion:

data: interval
linear relationship with no outlier and without multicollinearity

sample size: number of cases > number of factors

including relevant variables
true correlation between variables and factors

Exploratory factor analysis ≈ FA
Confirmatory factor analysis (CFA)

≈ PCA

Factor model (因子モデル)
Fixed (effect) model (固定モデル) = parametric model (母数モデル), FM: type I

The way to obtain common factors (共通因子)

• principal factor analysis (PFA) or principal axis factoring (PAF)
• centroid method
• varimax rotation for obtaining varimax solution

Discriminant analysis, DFA (判別分析)

= discriminant function analysis, DFA