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Stats, 2 Aggr, Ego Fallacy, Simp Para, ICC, Design Effect, 3 Nullmodell…

- - - - Hierarchies:
        
        ofc dep on Studiendesign/Fragestellung, the hierarchy can be either or (one over the other)
        
        Obere / Untere Ebene
        
        "Untere Ebene"
        
        "Level 1"
        
        "Micro Level"
        
        "Lower Level"
        
        lvl 1 Var sind abh innerhalb der lvl 2 Var
        
        "Obere Ebene"
        
        "Level 2"
        
        "Macro Level"
        
        "Upper Level"
        
        eg 3 level study design:
        Klasse => student => Messwdh
        
        Zusammenhänge im Multilevel Framework
        
        every var has an indiv value/ID assigned to them, not a shared value/ID
        
        thus each var is micro, and not macro (req shared values/IDs)
        
        Zeit
        is the micro Var
        
        there is a shared value "class size", meaning it is macro
        
        if the value is indiv, then its micro, eg grades
        
        eg: school to LK
        
        Left/right handed is a shared value ⇒ macro ⇐ doesnt change over Messwdh
        
        avg Augendruck is used, and is unique/indiv ⇒ micro ⇐ Messwiederholend, can change after every Messwdh
        
        eg: big 5 to religion
        
        Obere Ebene
        Untere Ebene
        
        Z = SES
        
        x = indiv verbale IQ
        
        y = indiv LK
        
        assigned Therapy grp ⇒ macro Z
        
        data collec multiple times ("longitudinal") ⇒ micro y
        
        Patients ⇒ micro x
- - - - for sure applies with longitudinal, since Zeit is always the micro level 1, and gender and personal data is macro level 2
  - - - so u have to aggr it first (Reading Score Level-2; sl 4)
      - or just create MLM (Reading Score Level-2; sl 3)
- - - - both micro vars
      - Zusammenhang zwischen Verantwortung und Arbeitszufriedenheit
        
        - ⇒ neg korrel ⇔ Je höher die Verantwortung, desto tiefer die Arbeitszufriedenheit.
        
        - ⇒ Simpson's Paradox
        
        Simpson‘s Paradox – Nichtberücksichtigung von Gruppen (Level-2) kann zu völlig anderen Resultaten führen.
        
        - "if you dont consider the upper grp levels, you get a diff correl line than if you do consider the grps"
        
        eg tog Streuung looks like a neg correl, but considering the grps you get a pos correl per grp - and the MLM will show the avg of the pos correls per grp
        
        why?:
        
        Die drei Firmen unterscheiden sich deutlich bzgl. der durchschnittlichen Verantwortung. (Verantw levels are diff btw the Firmen)
        
        wie auch der mittleren Arbeitszufriedenheit! (the Firmen means)
        
        again nicht Unabh, bc those in the same Firma have Abh!
        
        so u gotta make them Unabh then..
        
        by separating the Firmen. But now that you separate the Firmen tho it becomes the opp:
        
        - ⇒ pos korrel found! ⇔ Je höher die Verantwortung, desto höher die Arbeitszufriedenheit
  - - - ⇒ can and does work (typically similar results as MLM). but theres infoverlust.
        
        thats why u should use MLM to keep all info
        
        eg: theres diff avg Verantw levels for each Firma
        
        ⇒ abh exists btw Firma and Verantw
        
        eg Gruppengrössen diffs
        
        eg Detkoeff of lin Reg (prediction acc)
- - - - but now were looking at the avgs, meaning on another (upper) level.
        
        ⇒ no longer micro-micro, but macro-macro
        
        - ⇒ Ecological Fallacy
        
        we in turn answer a diff q. Not the q that was asked (namely per person - micro, and not per firma - macro)
  - - - Neighborhood
        
        Uptown
        - % College Graduates―80% - Average Income―$100,000
        
        Midtown
        - % College Graduates―40% - Average Income―$50,000
      - You might look at this and say:
        
        “People with more education earn more money.”
        
        But this is the ecological fallacy because you’re assuming that individuals with more education in the group are earning more, when all you actually know is that the group averages are correlated.
        
        👉 What if, within each neighborhood, the highest earners are actually those without degrees? Then your individual-level assumption would be wrong, even though the group data suggested otherwise.
        
        what if the 20% that arent graduated are the ones with the most income! (eg self made men)
        
        ⇒ its those who didnt graudate with high incomes that pull the avg income of the city up, not the college grads - altho the city graduate ratio makes it look that way!
  - - - ⇒ again the case of Ecological Fallacy
        
        it only applies to level 2, macro grp pro Gemeinde
        
        and not level 1, pro person
        
        meaning q is diff. not the more Einkommen per person but, the more Einkommen per Gemeinde ⇒ the more no's to 13. AHV
        
        ⇒ again need MLM to consider grp size, etc etc. Alleviating Eco Fallacy
- - - - since ICC very low ⇔ low abh (to which grp you belong ⇔ doesnt matter which grp/school you belong, the measurements are "fair") ⇔ "Unabh" ⇔ Unilevel models usable, MLM not necc
        
        low ICC ⇔ need more Messwdh to reach high Reliabilität
  - - - ICC high ⇔ high Abh (towards which grp/school you belong )⇔ the means are diff placed ⇔ need MLM
        
        high ICC ⇔ need less Messwdh to reach high Reliabilität
- - - - "Residuum" Rij = Abw of Yij from grp j mean
        
        = imprecision from the prediction line
        
        = residual error!
  - - - Note that Yijs Abh to the overall mean isnt a thing!
      - "Abh described in the ICC vanishes"
  - - - Unilevel models only describe the overall mean y00 and the var(Rij)
        => bc there are no grps U0j and thus no var(U0j), and thus no ICC, bc ICC describes the Var_zw Grp
      - Unilevel model var(Rij) ~= Nullmodell var(U0j) + var(Rij)
- - - - Kontexteffekt
        
        = lvl 2 coeff - lvl 1 coeff
        
        = grp mean coeff - indiv grp mean centered coeff
        
        | Effekt | Variable | Bedeutung |
        
        | ------------------ | ----------------------- | --------------------------------------------------- |
        
        | Level-1-Effekt | IQ_centered | Effekt des individuellen IQ im Vergleich zur Klasse |
        
        | Level-2-Effekt | mean_IQ | Einfluss des Klassen-IQ auf alle Schüler |
        
        | Kontexteffekt | mean_IQ – IQ_centered | Unterschied zwischen Gruppen- und Individualeffekt |
        
        eg:
        A negative context effect suggests that in higher-IQ schools, students with
        lower/average IQ actually do a bit worse than you'd expect from individual IQ alone — maybe due to competition or relative standing.
        => IQ++ schools => IQ-- students
    - - Grand Mean Centering = which meant to "lowkey standardise" the level 1 values by simply subtracting every level 1 value by the overall mean (= the mean of all means)
        
        𝑥ij grand mean centered
        
        => y01 = Kontexteffekt
        
        = Die Korrel Steigung zw IQ grp mean und der LK ist um 1.28 grösser als der Korrel zw dem indiv IQ und indiv LK
        
        => y01 = um welchen Betrag der Effekt des Level−2 Prädiktors auf die Zielvariable höher ist als jener des Level-1 Prädiktors
        
        red = Kontexteffekt
        = Steigungs diff
        = lvl 2 - lvl 1 Effekt or reg line Steigung
        
        green = lvl 2 effect
        
        black = lvl 1 effect
        
        ChatGPT
        
        eg:
        A negative context effect suggests that in higher-IQ schools, students with lower/average IQ actually do a bit worse than you'd expect from individual IQ alone — maybe due to competition or relative standing.
        -> mean IQ school++ => IQ students--
        
        A positive context effect means that being in a higher-performing group improves your outcome, even after accounting for your own individual characteristics.
        => You benefit just from being in a “better” context.
        -> mean IQ school++ => IQ students++
      - Group Mean Centering = as the name sugg, you subtract each level value 1, with its own grp level 2 mean,
        thus "lowkey standardising" that way
        
        xij group mean centered
        
        => y01 = RIM with lvl 2 Prädiktor Koeff
        
        => y01 = um welchen Betrag sich y ändert, wenn der Wert von ̅ 𝑥.j um eine Einheit zunimmt.
        
        => y01 = "Einfluss des Level−2 Prädiktors auf die Zielvariable"
      - y10 is the Steigung for level 1, in both centering methods it stays the same
        
        but y01 describes Steigung level 2, and thus changes dep on the type of centering
- - - - bc sport athletes are used to the intensity, thus their heart is calmer / the avg heart rate, the more sport they do is reduced
- - - - E.g., schools that start with higher scores improve faster.
    - - E.g., students who start strong don’t improve much.
  - - - ⇔ wenig Schwankung/Varianz zw den Gruppen/Schulen