CS188 Intro to AI Problem Set

Please see the problem set in attachment. Coding part is optional.Course material: https://inst.eecs.berkeley.edu/~cs188/fa20/

CS 188

Fall 2020

Introduction to

Artificial Intelligence Written HW 4

Due: Wednesday 12/02/2020 at 11:59pm (submit via Gradescope).

Policy: Can be solved in groups (acknowledge collaborators) but must be written up individually

Submission: Your submission should be a PDF that matches this template. Each page of the PDF should

align with the corresponding page of the template (page 1 has name/collaborators, question 1 begins on page

2, etc.). Do not reorder, split, combine, or add extra pages. The intention is that you print out the

template, write on the page in pen/pencil, and then scan or take pictures of the pages to make your submission.

You may also fill out this template digitally (e.g. using a tablet.)

First name

Last name

SID

Collaborators

For staff use only:

Q1. Probabilistic Language Modeling /40

Q2. Machine Learning /40

Q3. Optional: Programming Naive Bayes /0

Total /80

1

Q1. [40 pts] Probabilistic Language Modeling

In lecture, you saw an example of supervised learning where we used Naive Bayes for a binary classification problem:

To predict whether an email was ham or spam. To do so, we needed a labeled (i.e., ham or spam) dataset of emails.

To avoid this requirement for labeled datasets, let’s instead explore the area of unsupervised learning, where we don’t

need a labeled dataset. In this problem, let’s consider the setting of language modeling.

Language modeling is a field of Natural Language Processing (NLP) that tries to model the probability of the next

word, given the previous words. Here, instead of predicting a binary label of “yes” or “no,” we instead need to

predict a multiclass label, where the label is the word (from all possible words of the vocabulary) that is the correct

word for the blank that we want to fill in.

One possible way to model this problem is with Naive Bayes. Recall that in Naive Bayes, the features *X*1*, …, X**m*

are assumed to be pairwise independent when given the label *Y *. For this problem, let *Y *be the word we are trying

to predict, and our features be *X**i *for *i *= *−**n, …, **−*1*,*1*, …, n*, where *X**i *= *ith *word *i *places from *Y *. (For example,

*X*

*−*2 would be the word 2 places in front of *Y *. Again, recall that we assume each feature *X**i *to be independent of

each other, given the word *Y *. For example, in the sequence Neural networks ____ a lot, *X**−*2 = Neural, *X**−*1 =

networks, *Y *= the blank word (our label), *X*1 = a, and *X*2 = lot.

(a) First, let’s examine the problem of language modeling with Naive Bayes.

(i) [1 pt] Draw the Bayes Net structure for the Naive Bayes formulation of modeling the middle word of a

sequence given two preceding words and two succeeding words. You may think of the example sequence

listed above:

Neural networks ____ a lot.

(ii) [1 pt] Write the joint probability *P*(*X**−*2*, X**−*1*, Y, X*1*, X*2) in terms of the relevant Conditional Probability

Tables (CPTs) that describe the Bayes Net.

(iii) [1 pt] What is the size of the largest CPT involved in calculating the joint probability? Assume a

vocabulary size of *V *, so each variable can take on one of possible *V *values.

(iv) [1 pt] Write an expression of what label *y *that Naive Bayes would predict for *Y *(Hint: Your answer

should involve some kind of arg max and CPTs.)

(v) [3 pts] Describe 2 problems with the Naive Bayes Approach for the general problem of language modeling.

Hint: do you see any problems with the assumptions that this approach makes?

2

Now, let’s change our setting a bit. Instead of trying to fill in a blank given surrounding words, we are now only

given the preceding words. Say that we have a sequence of words: *X*1*, …, X**m**−*1*, X**m*. We know *{**X**i**}**m i*=0 *−*1 but we

don’t know *X*

*m*.

(b) For this part, assume that every word is conditioned on all previous words. We will call this the Sequence

Model.

(i) [1 pt] Draw the Bayes Net (of only *X*1*, X*2*, X*3*, X*4*, X*5) for a 5-word sequence, where we want to predict

the fifth word in a sequence *X*5 given the previous 4 words *X*1*, X*2*, X*3*, X*4. Again, we are assuming here

that each word depends on all previous words.

(ii) [1 pt] Write an expression for the joint distribution of a general sequence of length *m*: *P*(*X*1*, …, X**m*).

(iii) [1 pt] What is the size of the largest CPT involved in calculating the joint probability? Assume a

vocabulary size of *V *, so each variable can take on one of possible *V *values.

(c) You should have gotten a very large number for the previous part, which shows how infeasible the sequence

model is. Instead of the model above, let’s now examing another modeling option: N-grams. In N-gram

language modeling, we add back some conditional assumptions to bound the size of the CPTs that we consider.

We limit the tokens of consideration from “all previous words” to instead using only “the previous *N **−*1 words.”

This creates the conditional assumption that, given the previous *N **− *1 words, the current word is independent

of any word before the previous *N **− *1 words. For example, for *N *= 3, if we are trying to predict the 100th

word, then given the previous *N **− *1 = 2 words (98th and 99th words), then the 100th word is independent of

words 1*, . . . , *97 of the sequence.

(i) [1 pt] Making these additional conditional independence assumption changes our Bayes Net. Redraw the

Bayes Net from part (ci) to represent this new N-gram modeling of our 5-word sequence: *X*1*, X*2*, X*3*, X*4*, X*5.

Use *N *= 3.

3

(ii) [2 pts] Write an expression for the N-gram representation of the joint distribution of a general sequence

of length *m*: *P *(*X*1*, …, X**m*). Please use set notation (for example: For tokens *X**i**, …, X**j*, please write

something of the form *{**X**k**}**j k*=*i*). Your answer should express the joint distribution *P *(*{**X**i**}**m i*=1), in terms

of *m *and *N*.

Hint: If you find it helpful, try it for the 5 word graph above first before going to a general *m *length

sequence.

(iii) [1 pt] What is the size of the largest CPT involved in calculating the joint probability above? Again,

assume a vocabulary size of *V *, and *m > N*.

(iv) [2 pts] Describe one disadvantage of using N-gram over Naive Bayes.

(v) [4 pts] Describe an advantage and disadvantage of using N-gram over the Sequence Model above.

4

(d) In this question, we see a real-world application of smoothing in the context of language modeling.

Say we have the following training corpus from Ted Geisel:

i am sam . sam i am . i do not like green eggs and ham .

Consider the counts given in the tables below, as calculated from the sentence above.

1-gram

Token Count

i 3

am 2

sam 2

. 3

do 1

not 1

like 1

green 1

eggs 1

and 1

ham 1

TOTAL 17

2-gram phrases starting with i

Token1 Token2 Count

i am 2

i do 1

TOTAL 3

2-gram phrases starting with am

Token1 Token2 Count

am sam 1

am . 1

TOTAL 2

(i) [1 pt] Based on the above dataset and counts, what is the *N*-gram estimate for *N *= 1, for the sequence

of 3 tokens i am ham? In other words, what is *P*(*i, am, ham*) for *N *= 1?

(ii) [1 pt] Based on the above dataset and counts, what is the *N*-gram estimate for *N *= 2, for the sequence

of 3 tokens i am ham? In other words, what is *P*(*i, am, ham*) for *N *= 2?

(iii) [3 pts] What is the importance of smoothing in the context of language modeling?

Hint: see your answer for the previous subquestion.

5

(iv) [5 pts] Perform Laplace *k*-smoothing on the above problem and re-compute *P *(*i, am, ham*) with the

smoothed distribution, for *N *= 2. In order to calculate this, complete the pseudocount column for

each entry in the probability tables. Note we add a new <unk> entry, which represents any token not in

the table.

Hint: the count for the new <unk> row in each table would be 0.

1-gram

Token Count Pseudocount

i 3

am 2

sam 2

. 3

do 1

not 1

like 1

green 1

eggs 1

and 1

ham 1

*<*unk*> *0

TOTAL 17

2-gram phrases starting with “i”

Token1 Token2 Count Pseudocount

i am 2

i do 1

i *<*unk*> *0

TOTAL 3

2-gram phrases starting with “am”

Token1 Token2 Count Pseudocount

am sam 1

am . 1

am *<*unk*> *0

TOTAL 2

6

(v) [4 pts] What is a potential problem with Laplace smoothing? Propose a solution. (Assume that you have

chosen the best *k*, so finding the best *k *is not a problem.)

Hint: Consider the effect of smoothing on a small CPT.

(vi) [2 pts] Let the likelihood *L*(*k*) = *P *(*i, am, sam*), give an expression for the log likelihood ln *L*(*k*) of this

sequence after *k*-smoothing. Continue to assume *N *= 2.

(vii) [4 pts] Describe a procedure we could do to find a reasonable value of *k*. No mathematical computations

needed.

Hint: you might want to maximize the log likelihood ln *L*(*k*) on something.

7

Q2. [40 pts] Machine Learning

In this question, we will attempt to develop more intuition about how Neural Networks work. In parts A and B, we

will discuss gradient descent, and in part C we look at backprop.

(a) Gradient descent is a procedure which allows you to minimize any loss function. As as example, let’s consider

a simple function *Loss*(*w*) = *w*2 and let’s assume that we want to minimize this function. Perform gradient

descent on this loss function by using the update rule *w **← **w **− **α **dLoss dw *, where *α *is the learning rate.

(i) [1 pt] What is *dLoss dw *? Write your answer in terms of *w*.

(ii) [1 pt] What is the optimal *w *that minimizes this loss function? We denote this value of *w *as *w**∗*.

(iii) [2 pts] Carry out one iteration of gradient descent (i.e., weight update). What is the resulting weight

(and corresponding post-update loss) for the scenarios below? Plot the loss function (*w*2) by hand and,

for each of the two scenarios below, draw the direction in which *w *is updated (an arrow on the *w *axis

from *w**old *to *w**new*).

1. *α *= 0*.*1, *w *= 2

2. *α *= 1, *w *= *−*2

8

(iv) [4 pts] Assume *w *is initialized to some nonzero value. Assume we are still working with *Loss*(*w*) = *w*2

1. Which value of *α *allows gradient descent to make *w *converge to *w**∗ *in the least amount of steps?

2. For what range of *α *does *w *never converge?

Hint: You may consider for which *γ *where *w**t *= *γw**t**−*1 will never converge.

(v) [2 pts] Why must *α *always be positive when performing gradient descent?

(b) It is unlikely that we have a loss function as nice as *w*2. Say we instead want to minimize some more complex

loss *Loss*(*w*) = *w *24 + *w *33 *− *3*w*2 + 7, a polynomial with local minima at *w *= *−*2*, *1*.*5, a global minimum at

*w *= *−*2, a local maximum at *w *= 0, and limits that go to infinity for both *w **→ ∞ *and *w **→ −∞*.

*−*3 *−*2 *−*1 0 1 2 3

0

10

20

30

*w*

*Loss*(*w*)

9

(i) [3 pts] Why do Neural Networks use gradient descent for optimization instead of just taking the derivative

and setting it equal to 0? Explain in 1-2 sentences. You may use the example error function from above

to explain your reasoning.

(ii) [1 pt] What is the optimal *w**∗*, given the loss above?

(iii) [3 pts] Let *α *and *w *take on the values below. For each case, perform some update steps and report

whether or not gradient descent will converge to the optimum *w**∗ *after an infinite number of steps. If not,

report whether it converges to some other value, or does not converge to any value.

1. *α *= 1, *w *= 0

2. *α *= 1, *w *= *−*2

3. *α *= 1, *w *= 1

4. *α *= 0*.*1*, w *= 3

10

5. *α *= 0*.*1*, w *= 2

6. *α *= 0*.*1*, w *= *−*10

(iv) [3 pts] From the subquestion above, explain in 1-2 sentences the effect of learning rate being (a) too high

and (b) too low.

(c) Let’s now look at some basic neural networks drawn as computation graphs.

(i) [1 pt] Consider the following computation graph, which represents a 1 layer Neural Network.

*x m*

*× **b*

+ *f*(*x*)

1. Write the equation for the network’s output (*y *= *f*(*x*)) in terms of *m, x, b*.

2. Describe the types of functions that can be encoded by such a function (given that the variables that

it can control are *m *and *b*).

11

(ii) [2 pts] Let’s stack two of the above graphs together to (potentially) represent a 2 layer “Neural Network.”

*x m*

1

*× **b*1

+

*×*

*m*2 *b*2

+ *f*(*x*)

1. Write the equation for *y *= *f*(*x*) in terms of *m*1*, m*2*, b*1*, b*2*, x*.

2. Describe the types of functions that can be encoded by such a function (given that the variables that

it can control are the 4 learnable parameters *m*1*, m*2*, b*1*, b*2). Compare this with the previous neural

network’s expressive power.

3. Is this actually a 2-layer network? If it is, explain in 1-2 sentences. If not, rewrite it (algebraically)

as a 1-layer network with only 2 learnable weights. Why do neural networks need nonlinearities?

(iii) [4 pts] Now, let’s go back to the first NN and add a nonlinearity node. Recall *ReLU*(*x*) = max (0*, x*).

Also consider a loss function *Loss*(*y, y**∗*) which represents the error between our network’s output (*y*) and

the true label (*y**∗*) from the dataset. We will perform an abbreviated version of backpropagation on this

network.

*x m*

*×*

*a*

*b*

+ ReLU

*y*

Loss

*y**∗*

1. Compute *∂Loss ∂a *using Chain Rule. Use Mean Squared Error as the loss function, which is defined as

*MSE*(*y, y**∗*) = (*y **−**y**∗*)2 where *y**∗ *= true label and *y *is the predicted output from the neural network.

2. Find *∂Loss*

*∂b *. Note that since we are doing backprop, we can reuse calculations from part 1.

12

3. Find *∂Loss*

*∂m *. Note that since we are doing backprop, we can reuse calculations from part 1.

4. What is the gradient descent update rule for updating *m*? What is the update rule for *b*?

For the next few parts, we analyze the Perceptron algorithm. In the perceptron algorithm, we predict +1 if *~w**T **f ~*(*x*) *≥*

0, and predict *−*1 else, where *f ~*(*x*) is a feature vector.

(d) [3 pts] When implementing the perceptron algorithm with a neural network, the following function might be

of use: *sign*(*x*) = (1 if *−*1 if *x x < **≥ *0 0. If we added this *sign*(*x*) node to our neural network drawings, what would

happen during backpropagation through this node?

Hint: what does the gradient look like for various *x *values?

13

(e) [2 pts] Draw the binary perceptron prediction function as a “neural-network”-styled computation graph. Assume 3 dimensional weight and feature vectors: that is, [*w*0*, w*1*, w*2] is the weight vector and [*f*0(*x*)*, f*1(*x*)*, f*2(*x*)]

is the feature vector. Recall that in the perceptron algorithm, we take the dot product of the weight vector

and the feature vector. In addition to the addition and multiplication nodes, add a loss node at the end, to

represent the prediction error which we would like to minimize. Label the edge which represents the perceptron

model’s output as *y*.

Hint: *y *= *sign*(*w*0 *∗ **f*0(*x*) + *w*1 *∗ **f*1(*x*) + *w*2 *∗ **f*2(*x*))

(f) [2 pts] Using Mean Squared Error (*y **− **y**∗*)2 as the loss function, compute *∂Loss ∂w**i *. Because of the problem you

noticed in the previous part with including the *sign *node, as we are doing chain rule below, use the custom

gradient *∂sign ∂x*(*x*) = h *∂sign ∂x*(*x*) i*custom *= 1.

14

(g) In this part, we will derive the gradient update rule for the perceptron using our graph above.

(i) [1 pt] The loss gradient is defined as *∇**w**Loss *=

*∂Loss*

*∂w*0

*∂Loss*

*∂w*1

*∂Loss*

*∂w*2

. Using your answer from the previous question,

write out the loss gradient.

(ii) [4 pts] What is the gradient update rule ( *~w **← **~w **− **α**∇**w**Loss*) for the cases below?

Hint: your answers will be in terms of *f ~*(*x*) and *α*.

1. *y *= *−*1*, y**∗ *= 1

2. *y *= 1*, y**∗ *= *−*1

3. *y *= *y**∗*

(iii) [1 pt] For *α *= 1 4, compare the update rules you derived for the 3 cases above with the Perceptron update

formula in the notes and lecture. Briefly describe your observations.

15

Q3. [0 pts] Optional: Programming Naive Bayes

Now we will implement these ideas in code (in a Google Colab Notebook, link posted on piazza) to perform language

modeling, and then use the language model to generate some novel text!

(a) You will implement some of the math you computed earlier to complete the following functions in the provided

N-gram class. Note that if you follow our hints, this should not require more than 15 total lines of code for all

the functions below.

First, follow the instructions in the instruction PDF to set up your Google Colab Notebook.

You can find the pdf under

Piazza/resource/CS 188 Fall 2020 Written Homework 4 Colab Instructions (Optional).pdf

You will need to implement the following functions:

• count_words

1. This function returns a dictionary with the count of each word in self.text_list.

2. HINT: You can do this in one line by using collections.Counter.

• calc_word_probs

1. This function converts a dictionary of counts from count_words and normalizes the counts into

probabilities.

2. HINT: You can do this in 1-2 lines by using self.normalize_dict(…)

• probs_to_neg_log_probs

1. This function converts an inputted dictionary of probabilities probs_dict into a dictionary of negative

log probabilities.

2. HINT: Use np.log.

• filter_adj_counter

1. This function is a little more complicated. Given a length *N **− *1 tuple of tokens and their associated counts (frequencies), this function searches through all the length *N *phrases it has stored in

self.adj_counter (or is passed in via the argument adj_counter) and returns a dictionary with

only the length *N *phrases with the same first *N **−*1 words as word_tuple, plus their associated counts

(frequencies). See the docstring for a concrete example.

2. HINT1: Use phrase[:len(word_tuple)] to get a tuple of the first *N **− *1 words of each *N*-length

phrase in the adj_counter to compare with word_tuple.

3. HINT2: We are returning the filtered dictionary which is stored in the variable subset_word_adj_counter,

so you need to modify this dictionary in some way.

• p_naive

1. This function calculates the non-smoothed empirical probability of a length *n *phrase occurring given

length *n**−*1 tuple of tokens prev_phrase. In other words, it calculates *P*(current token*|*previous N – 1 tokens).

The probability is based on counts, exactly like how we calculated probabilities in the green eggs and

ham example earlier in this problem without smoothing.

2. HINT1: You need to define prob because it is being returned.

3. HINT2: You need to normalize filtered_adj_counter which is already defined for you.

• calc_neg_log_prob_of_sentence

1. This function calculates and returns the negative log probability of the entire sequence of tokens

sentence_list given a probability function p_func (which is either the smoothed or the nonsmoothed probabilities).

2. HINT1: curr_word_prob is defined for you, and is *P*(currToken*|*previous N – 1 tokens).

3. HINT2: cum_neg_log_prob is what the function returns. For each iteration of the for-loop, what

must we do to update cum_neg_log_prob?

4. HINT3: Think about how we combine log probabilities for each word.

• calc_prob_of_sentence

1. This function calculates and returns the probability of a sequence of tokens.

16

2. HINT1: Use the function you just wrote, calc_neg_log_prob_of_sentence.

3. HINT2: Use np.exp.

(b) After writing the above functions and mounting the corpus on your google drive, you should be able to run the

text generation algorithm. This algorithm works by first using the N-gram model to construct CPTs (as we

saw earlier in this homework). Then, it uses the CPTs to generate a sequence of words that our model thinks

can occur with relatively “high probability.” Our hope is that the “high probability” sequences are sequences

of words that make some kind of sense.

Run the text generation algorithm and record (in the spaces below) some of your N-gram model-generated

sentences with the following parameters. Please do these in order or else you will be very disappointed by the

mediocrity of the text generated. What are the effects of increasing *N *and *k *on the quality of the generated

text? Modify the *N, k *variables in the “play with params in code here” section.

1. *N *= 1, *k *= 1

2. *N *= 2, *k *= 1

3. *N *= 2, *k *= 5

4. *N *= 3, *k *= 1

5. *N *= 3, *k *= 5

(c) Now, perform language modeling on a dataset/corpus of your choosing. Select a corpus, put it in a text

file, and upload it to the google drive folder with all the other .txt files you uploaded earlier. Then redefine

training_corpora_path_list under the comment “REDEFINE training_corpora_path_list here if you

wish to use your own corpus” and use the model to perform text generation (as you did above). For good

results, select a corpus at least 50,000 words long. If you are not feeling creative, feel free to use the other files

in the cs188whw4public folder.

Below, write a sentence that your N-gram model generated on your custom corpus.

(d) So far, we have used N-gram to do language modeling and text generation. We can also use N-gram, with some

modifications (that staff has already coded for you) to capitalize a sequence of words correctly. This is done

using probability maximization; in other words, which options of capitalization look most like things we have

seen in the training dataset. The current implementation is slow, but if you were to make a spin-off project,

you can look into the Viterbi Algorithm (not in scope for 188 this semester) to speed it up.

Run the capitalization script on the strings provided (you do not need to make any changes for this part). If

you wrote your code correctly, you should get capitalizations of the inputted sentences that make sense. In

other words, your model is smart enough to know when to capitalize tricky words like “united!”

17

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Laralex Case Study Data | ||||||||||

Hospital Acquired Infections | Cesarean Section Procedures | Discrepant X-Rays | Unscheduled Readmissions | Patients Who Leave the ED Prior to Treatment | ||||||

Month | Patient-Days | No. | Births | No. | Patients | No. | Patients | No. | Patients | No. |

1 | 5225 | 22 | 119 | 32 | 488 | 8 | 310 | 6 | 604 | 6 |

2 | 5515 | 20 | 111 | 27 | 573 | 3 | 294 | 5 | 575 | 10 |

3 | 5872 | 15 | 111 | 32 | 489 | 6 | 337 | 14 | 593 | 7 |

4 | 5398 | 22 | 125 | 28 | 420 | 4 | 253 | 10 | 641 | 6 |

5 | 5017 | 26 | 99 | 27 | 503 | 6 | 293 | 10 | 601 | 11 |

6 | 5273 | 17 | 127 | 27 | 580 | 7 | 300 | 4 | 649 | 9 |

7 | 4824 | 20 | 121 | 25 | 419 | 8 | 319 | 10 | 658 | 11 |

8 | 5340 | 21 | 117 | 32 | 442 | 4 | 199 | 7 | 552 | 11 |

9 | 5307 | 14 | 133 | 30 | 407 | 3 | 263 | 11 | 536 | 9 |

10 | 5507 | 20 | 106 | 23 | 553 | 9 | 259 | 5 | 554 | 11 |

11 | 4189 | 22 | 120 | 27 | 466 | 3 | 285 | 14 | 708 | 11 |

12 | 4378 | 17 | 123 | 33 | 551 | 4 | 275 | 11 | 547 | 12 |

13 | 4620 | 20 | 114 | 29 | 485 | 10 | 320 | 13 | 589 | 16 |

14 | 5869 | 27 | 128 | 19 | 427 | 7 | 329 | 12 | 596 | 12 |

15 | 4975 | 21 | 117 | 19 | 540 | 9 | 243 | 11 | 685 | 18 |

16 | 4969 | 19 | 115 | 21 | 568 | 3 | 278 | 8 | 640 | 15 |

17 | 5792 | 17 | 104 | 22 | 531 | 9 | 365 | 6 | 659 | 17 |

18 | 4939 | 22 | 128 | 20 | 558 | 5 | 348 | 11 | 609 | 16 |

19 | 5616 | 16 | 120 | 24 | 474 | 4 | 290 | 8 | 438 | 14 |

20 | 5061 | 11 | 121 | 25 | 594 | 9 | 321 | 7 | 522 | 13 |

21 | 5262 | 20 | 102 | 21 | 540 | 2 | 253 | 9 | 574 | 16 |

22 | 4808 | 26 | 107 | 18 | 553 | 9 | 266 | 10 | 539 | 18 |

23 | 5280 | 20 | 118 | 24 | 556 | 11 | 301 | 11 | 634 | 21 |

24 | 5491 | 24 | 116 | 22 | 541 | 7 | 348 | 9 | 610 | 22 |

BOSTON

UNIVERSITY

METROPOLITAN COLLEGE

DEPARTMENT OF ADMINISTRATIVE SCIENCES

LARALEX HOSPITAL1

Blanche Davis just completed her third year as Director of Quality at Laralex Hospital, a medium-sized notfor-profit facility located in a growing region of the Southeastern United States. Laralex Hospital, a member

of the Southeast Medical Care Group, offers a wide range of services to patients who typically belong to one

of the three major managed care providers in the area. The 260-bed facility includes numerous departments

such as maternity, emergency, cardiac care, diagnostic testing, and medical imaging.

Blanche’s primary responsibility is maintaining the Hospital’s accreditation status, which the Board of

Directors considers critical to the Hospital’s long-term viability. In order to maintain accreditation, a

hospital must submit to audits, both through written documentation and on-site visits, designed to evaluate

its operations against recognized best practices. Hospitals must also provide the agency with periodic

updates, including routine ongoing performance data. Flexibility exists relative to the procedures used at

individual hospitals to evaluate performance.

Many hospitals use external performance benchmarking systems. These systems are administered by

independent organizations that collect data from participating hospitals, then place each facility into a peer

group of similar facilities, so that a comparison may be made. The organization used by Laralex Hospital

charges $12,500 per year for their service. In addition, Laralex employs another firm that analyzes data from

patient satisfaction surveys. Table 1 includes some of the performance data collected at Laralex Hospital.

Neonatal Mortality Rate

Hospital-Acquired Infections Rate

Surgical Wound Infections Rate

Inpatient Mortality Rate

Diagnostic Testing False Positive Rate

Patient Satisfaction Rate (Based on Surveys)

Cesarean Section Birth Rate

Rate of Patients Who Leave Emergency Department Prior to Service

Rate of Unscheduled Readmissions to the Hospital

Rate of Positive/Negative HIV, Hepatitis and Other Laboratory Results

Biopsy Results (Positive/Negative)

Medication Error Rate

Discrepant X-Ray Report Rate

Rate of Pap Smear Results by Category

Blanche has worked at Laralex Hospital for 24 years, ever since completing her education and becoming a

registered nurse. Having held a variety of professional and administrative positions in the Hospital, she is

well respected for her understanding of all internal operations. One morning as Blanche arrived for work,

she found the most recent quarterly benchmark analysis, which compared the Laralex’s performance data to

its peer group of hospitals. There was also a voice mail message from Hazel Wisely, Vice President of

Quality Assurance and Risk Management. “Blanche, take a look at the latest benchmark report. Our results

1 This case was developed by John Maleyeff and F.C. Kaminsky based on their work in applying quality management

principles in healthcare settings. All references to people and organizations are fictional. © 2020 (Rev) All rights reserved.

Laralex Hospital Case Study Page 2

for hospital-acquired infections, x-ray report discrepancies, and unscheduled readmissions are way up. I am

especially concerned about the increase in hospital-acquired infections. What’s going on?” Blanche opened

the report and found that the rate for hospital-acquired infections (an infection that a patient experiences that

was not present at the time of admission) was 4.5 per 1000 patient-days and the corresponding percentile

ranking (compared to the other hospitals in Laralex’s peer group) was 86. This percentile means that the

infection rate at Laralex was higher than 86% of the peer group hospitals.

The hospital-acquired infection rate was highlighted because, in the previous quarter, the infection rate was

only 2.9 per 1000 patient-days and the percentile ranking was 22. Blanche knew that these infections could

be due to many causes within almost any department in the hospital, such as personnel not washing their

hands according to the hospital’s protocol, not properly sterilizing treatment devices, or allowing patients to

move unattended around the hospital, to name a few. The latest benchmarking report also showed similar

results for x-ray discrepancies (a jump from 12 to 68 in percentile ranking) and unscheduled readmissions

(an increase from 32 to 91 in percentile ranking). Blanche knew that x-ray discrepancies could be caused by

patients not being instructed properly, improper use of the x-ray device, or device malfunctions, to name a

few. She also knew that unscheduled readmissions had many potential causes.

These types of requests were not new to Blanche. She generally received them whenever a quarterly report

comparing Laralex with other hospitals was generated. In response to these requests, Blanche would make a

few calls and visit the departments responsible for each performance measure. Typically, the department

manager’s first response would be similar to that of Bill Karinsky who managed the x-ray department and

was Blanche’s first stop. “As far as I know, we haven’t made any changes that would impact discrepancies,

but I’ll take a look.” If the interaction proceeded in a typical manner, Bill would talk to his technicians and

get back to Blanche with his best guess as to the reason for the increase. In most cases, the data from the next

quarterly performance benchmark report would show an improvement, and the issue would be forgotten.

Two aspects of the benchmarking system have continued to disturbed Blanche. First, the number of requests

to track down reasons for performance problems consumed a significant portion of her time. The frequency

of these requests seems to be unchanged over the last three years. Second, rarely was a definitive root cause

identified. The long-term data appears to indicate no real improvements in the hospital’s performance. On

this day, she was too busy to worry about these issues, because she needed to meet with the managers

responsible for the two other performance measures whose percentile rankings had slipped. Then, she was

off to a one-week training program on Six Sigma and she needs to start packing. She also needs to arrange

for a friend to feed her cat and water her garden plants.

Ever since the Six Sigma training course ended, Blanche could not stop thinking about something said by the

head trainer, Professor Robert Cavanaugh. When recommending procedures for interpreting the meaning of

performance data, Professor Cavanaugh stressed that

thought passed for a moment, and then she realized that she was thinking about the tomatoes on her 12

plants. In particular, she had 12 plants that were rooted in the same soil. They came from the same seed

packet, they were planted by the same gardener, and they were maintained in the same manner. The plants

are produced tomatoes in essentially equal amounts, both in size and quantity. In addition, the occasional

“bad” tomatoes seemed to occur uniformly across the 12 plants. Yet, individual tomatoes picked from a

plant would exhibit quite significant size variation. As she periodically picked the bad tomatoes from the

plants each Saturday, the number of both good and bad tomatoes picked from an individual plant varied

from Saturday-to-Saturday. Did this mean the outcomes changed (e.g., the size and number of bad

tomatoes) while the process remained the same?

Blanche came to realize that plants were like hospitals and tomatoes were like patients. The 12 plants would

be analogous to 12 hospitals in a peer group that were all managed identically and served similar

populations. Even though the 12 hospitals could be essentially the same, the occurrence of outcomes (such

as hospital-acquired infections) would differ across hospitals over a period (such as one quarter). Professor

Cavanaugh referred to these differences as random variations. Performance data for identical hospitals will

Laralex Hospital Case Study Page 3

vary both over time within each hospital and from hospital-to-hospital for the same period (just like the

occurrence of bad tomatoes on the 12 plants). If this analogy were accurate, then one hospital’s performance

within a group of peers could just as likely be the minimum of the peer group, or the maximum of the peer

group, or any place in the middle. Could this mean that the percentile ranking could vary from zero to 100

with equal likelihood? If so, then a percentile ranking change from 22 to 86 (which occurred for hospitalacquired infections) may mean nothing at all!

Blanche arrived early for work on the following Monday. The first thing she did was make coffee, since she

arrived before any of her support staff and was anxious to get to work exploring the performance data.

Using her limited database management skill, she accessed the raw data for hospital-acquired infections over

the past two years. For reporting purposes, the data had been recorded using monthly time intervals and

then summarized by quarter. To save time, Blanche used the monthly performance values. She quickly

downloaded the data (number of infections and number of patient-days by month, typically around 5,000)

and calculated each month’s infection rate over the 2-year period. After choosing the time series graph

option and clicking the OK button, she knew her hunch was confirmed. The “run chart” she created is

shown in Figure 1.

Blanche recognized the pattern shown on the run chart for hospital-acquired infections. The graph was

similar to many examples she saw during her Six Sigma training – it appeared to be a display of random

variation. She noticed that the run chart showed an average infection rate of about 0.4% (4 infections per

1000 patient-days) with monthly values varying around this constant average. In fact, for a given month, it

looked as though one could anticipate the rate to vary from about 2 per 1000 to about 6 per 1000. Could she

be looking at a process that is unchanged, with the data changing merely due to random variation? Or, was

the likelihood of an infection actually changing in the hospital over that period? Blanche left for a midmorning staff meeting.

On Tuesday, Blanche accessed the database containing performance data over the last two years. She

created run charts for five performance outcomes, including the three outcomes that she was asked about

last month. Figures 2-5 contain the run charts for the proportion of births having a Cesarean Section (CSection) procedure, the proportion of discrepant x-ray reports, the proportion of unscheduled readmissions

to the hospital, and the proportion of patients who left the emergency department (ED) prior to treatment. It

seemed to Blanche that random patterns of variation existed for all but two of the performance outcomes. In

the case of C-Section births (Figure 2), there appeared to be a change in the process (the proportions seemed

to drop suddenly, then maintain a consistent level). In the case of patients who left the ED prior to treatment

(Figure 5), there appeared to be a steady increase over time. Neither of these outcomes had previously been

highlighted using the peer group comparisons based on percentiles. The other outcomes, including hospitalacquired infections, seem to show random variation with no process changes.

2 4 6 8 10 12 14 16 18 20 22 24

0.0055

0.0050

0.0045

0.0040

0.0035

0.0030

0.0025

0.0020

Month

Proportion

Hospital Acquired Infections

Laralex Hospital Case Study Page 4

2 4 6 8 10 12 14 16 18 20 22 24

0.300

0.275

0.250

0.225

0.200

0.175

0.150

Month

Proportion

C-Section Births

2 4 6 8 10 12 14 16 18 20 22 24

0.0225

0.0200

0.0175

0.0150

0.0125

0.0100

0.0075

0.0050

Month

Proportion

Discrepant X-rays

2 4 6 8 10 12 14 16 18 20 22 24

0.05

0.04

0.03

0.02

0.01

Month

Proportion

Unscheduled Readmissions

Laralex Hospital Case Study Page 5

Blanche took a walk. Her first destination was the maternity department. Before she could say a word, the

head nurse, Robin Gallagher, who was never happy to see Blanche, began to provide a long list of reasons

why her C-Section numbers were bad. They included physicians scheduling the procedures in order to take

vacations, improper pre-natal care, or flu outbreaks potentially affecting pregnancies, among others.

However, this visit was different. After Blanche showed her the run chart for C-Sections, the Robin

immediately stated, “Sure, that’s when Dr. Forster left. Some of us thought that he was using the C-Section

procedure even in cases where the other doctors would not.” Blanche was thrilled because, finally, she

received a seemingly valid reason from the usually uncooperative Robin.

Blanche also visited the emergency department. After viewing the run chart of patients who left the ED

prior to treatment, the ED staff manager explained that the chart confirmed his impression that service in the

ED had gradually declined due to both an increase in patient demand as well as a decrease in the ED budget.

He had previously attempted to have his budget increased but was told that the performance benchmark

analysis never indicated a problem. He asked Blanche for a copy of the run chart for his use in justifying

future requests. At this point, Blanche realized how much time she had likely wasted over the last three

years tracking down problems that did not exist, and the lost opportunity to highlight real problems that the

current quality system was not able to identify.

Back in her office, Blanche reviewed the materials presented in the Six Sigma training program. Although

run charts can be effective at determining if a process has changed, statistical control charts provide a more

statistically sophisticated method. Although the details were not fresh in her mind, she did understand the

use of control charts during the training course. She plans to review the coverage of control charts and the

method presented for using statistical confidence intervals to determine if the outcomes of a process are

consistent with external benchmarks. In the meantime, she had some internal challenges to overcome.

Blanche Davis was able to convince Hazel Wisely and the leadership of Laralex that implementing a processoriented approach to analyzing quality-related data was consistent with accreditation requirements and

positioned the Hospital well with insurers, government regulators, and other key stakeholders. In fact, one

large insurer recently expressed concern to Laralex President Ingrid Carney about the similarity in quality

between accredited and non-accredited hospitals. This insurer may be instituting internal standards against

which to evaluate hospitals based specifically on their ability to improve over time. Ingrid knows that, by

creating an effective system now, these standards may in fact be based on the system implemented at

Laralex. This result would be a huge competitive advantage for the Hospital.

Blanche is given responsibility for organizing the new quality system, which will include the use of P Charts

and other statistical process control methods. It would monitor performance over time and provide a

statistically valid mechanism for comparison with peer hospitals. The new system would satisfy Joint

2 4 6 8 10 12 14 16 18 20 22 24

0.035

0.030

0.025

0.020

0.015

0.010

Month

Proportion

Left ED Prior to Treatment

Laralex Hospital Case Study Page 6

Commission requirements that performance metrics be evaluated both internally (against prior performance)

and externally (against suitable benchmarks). Its forward-looking approach that reacts quickly to unstable

processes also satisfies the requirement that performance data is used to initiate action to find root causes of

problems.

Blanche will also oversee the implementation of a process improvement system based on the aspects of Lean

Six Sigma that make sense within the Hospital’s environment. Laralex is unique because of its mix of

technically sophisticated medical staff and non-technical support staff. However, its technically oriented

staff (physicians, physiatrists, registered nurses, and medical imaging technicians) do not possess a strong

background in statistics. The hospital’s support functions (accounting, human relations, information

technology, purchasing, etc.) includes intelligent employees but many of whom are not mathematically

oriented.

Blanche’s main worry is that the culture within the Hospital may be inconsistent with the culture needed to

implement an effective process improvement program. At a recent management seminar, she heard the

phrase

modifications to their organizational infrastructure, but she does not know how to approach this aspect of

the implementation. She is especially concerned about the nurses, who are represented by a regional nurses

union, the Amalgamated Nurse and Midwife Union (ANMU), which recently had contentious relationship

with the Hospital during the recent round of contract negotiations. In the past, the ANMU did cooperate

with hospital leadership regarding working rules, but only if it was confident that the changes also

benefitted the nursing staff.

Blanche remembers a recent study performed by her staff showing that the rate of prescription errors was 2.8

per 100 prescriptions. Although most of these errors were minor (e.g., the date was not entered), this error

rate was a surprise to the Hospital’s leadership. As a result, the pharmacy manager was fired. Blanche

knows that, with a healthy process improvement program, all workers will need to report mistakes and close

calls, even if they personally made or almost made the mistake. For example, she wonders if nurses will

report these situations because they are often the last step in a complicated medication administration

process, and the last step is often blamed for problems.

She is also concerned that outside stakeholders may not understand that reporting problems in no way

implies that the Hospital’s operations are problematic. In fact, the U.S. Federal Aviation Administration

operates a voluntary system for tabulating and reporting mistakes and close calls based on information

provided by pilots, air traffic controllers, flight attendants, and other airline workers and contractors. Those

reporting incidents can do so anonymously and the reporter is immune from punishment. The success of

this system is evidenced by the impressive safety record of major U.S. airlines. Other industries, especially

healthcare, have taken strives to create similar systems. But, other leaders within the healthcare community

believe that the culture of litigation in the U.S. will prevent hospitals from willingly exposing its problems.

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