I thought of the following math problem:
Suppose there is a basketball coach that wants to test the following hypothesis: The coach believes that once a player successfully scores a few baskets - the player is then more likely to score more baskets.
Suppose the basketball coach then carries out an experiment - the coach asks different players to shoot baskets and records the results. As an example, the data might look something like this (I simulated this using the R programming language):
Player Baskets 1 Player 1 Miss Miss Miss Miss Hit Hit Hit Miss Miss Hit 2 Player 2 Hit Miss Miss Hit Hit 3 Player 3 Miss Miss Hit Hit Hit Miss Hit Miss Hit Miss Hit Hit 4 Player 4 Hit Hit Hit Miss Miss Hit Miss Miss Miss Hit Hit Miss Hit Miss 5 Player 5 Hit Hit Miss Hit Miss Miss Hit Miss
Based on this format of data - I thought of the 3 following methods to answer the coach's question:
Method 1: For each individual player, count the number of times (in general, over all shots) that "Miss" follows "Miss", "Miss" follows "Hit", "Hit" follows "Hit" and "Hit" follows "Hit". Repeat this for all players, and then you can construct a 4-State Markov Chain with Conditional Probabilities
Method 2: For each individual player, ignore everything except the last two shots. Then, count the number of at "Miss" follows "Miss", "Miss" follows "Hit", "Hit" follows "Hit" and "Hit" follows "Hit". Repeat this for all players, and then you can construct a 4-State Markov Chain with Conditional Probabilities only taking into consideration the last shot.
Method 3: For each individual player, assign a value of "1" when a basket is "Hit" and a value of "0" when a basket is missed". Then calculate the average for each player (e.g. Hit, Hit, Miss, Hit, Hit = 1+1+0+1+1 / 5 = 0.8) but ignore the last basket.
The data should now be in the following format:
Player Average_of_All_Baskets_Excluding_Last Last_Basket 1 Player 1 0.330 Hit 2 Player 2 0.500 Hit 3 Player 3 0.545 Hit 4 Player 4 0.500 Miss 5 Player 5 0.570 Miss
Based on this approach, a Regression Model (e.g. Logistic Regression) can be fit that models the probability of making the next basket, based on the scoring average of the player up until that point. We can also add other variables such as the "result of the second last basket" or the "average of the second last and the third last baskets" that can try to capture and benefit the model with more recent information. As a result, this model will try to estimate the conditional probability of making the next basket based on the history of the player. However, I do not know if using "lagged values of the response variable" will violate the assumptions of the Regression Model.
Thus, my question is - are all 3 methods valid approaches to estimating the conditional probabilities and testing the hypothesis of the coach? Are some of these methodologies more "valid" than others (e.g. perhaps some contain inherent biases and flaws)?